There are a few indications of engagement with volume during prayer in the Tanakh itself. Most prominent is a fairly detailed description of Hannah's prayer for a son in I Shmuel 1:12-13:
וְהָיָה כִּי הִרְבְּתָה לְהִתְפַּלֵּל לִפְנֵי יְקֹוָק וְעֵלִי שֹׁמֵר אֶת פִּיהָ:
וְחַנָּה הִיא מְדַבֶּרֶת עַל לִבָּהּ רַק שְׂפָתֶיהָ נָּעוֹת וְקוֹלָהּ לֹא יִשָּׁמֵעַ וַיַּחְשְׁבֶהָ עֵלִי לְשִׁכֹּרָה:
We are told that Hannah prayed at length and she was "speaking to herself, only her lips moved but her voice could not be heard." Apparently, this was not a normal protocol for prayer, since Eli, the priest who was watching her, thought her to be drunk.
Other models include the image in I Melakhim 18:27 of the prophets of Ba'al, who, in their showdown with Eliyahu on Mount Carmel, call out to Ba'al--with Eliyahu's urging--"in a loud voice." The Gentile king of the city of Nineveh calls on his people to repent in order to avoid the city's destruction at God's hand. As part of the protocol of repentance, he urges the people to "call out to God with force" in Yonah 3:8. A similarly loud prayer is described at a national assembly of the returning Exiles in Nehemiah 9:4.
As we enter into the rabbinic period, the details of Hannah's prayer achieve great prominence. The specific concern of not praying too loudly, in the context of the Amidah, is already attested in Tosefta Berakhot 3:6. The Tosefta asks, "Could it be that one should make one's voice heard? The story about Hannah is explicit: 'She spoke to herself.'" On Talmud Bavli Berakhot 31a, a similar tradition emphasizes the part of the verse that says "her voice was not heard" as support for the idea that it is forbidden to raise one's voice in prayer. A baraita on Talmud Bavli Berakhot 24b states that one who makes one's voice heard during the Amidah is of little faith (as it implies that added volume is required for God to hear prayer) and one who raises one's voice in prayer is like a false prophet (perhaps evoking the negative image of the shouting prophets of Ba'al).
The formulations here are somewhat ambiguous as to whether they insist that one should not make any sound, or whether one should simply be sure not to be excessively loud. [Indeed, a focus on prayer not being demonstratively public, as opposed to needing to be silent, seems to be the concern in a number of passages in the Christian Bible as well.]
One strand took a very hard stance on virtual silence during the Amidah, seeing the negative aspects of loud prayer as extending even to saying the words such that one would hear them oneself. Tanna D'vei Eliyahu 26 (ed. Friedmann), in a clear expansion of the traditions on Bavli Berakhot 24b, states that one who prays loudly enough to hear oneself gives false testimony and/or is of little faith. This implies that prayer should be totally silent. [Some versions of the Tosefta above end up getting influenced by this strand and forbid prayer that reaches even one's own ears.] This view lives on well into the middle ages and is cited by Rashba, Meiri and others. Most prominently, Zohar Vayakhel 202a seems to suggest that a prayer heard by any human ear (possibly including the ear of the pray-er himself) cannot fully be heard in the heavens.
Another strand seems to be more permissive and possibly even encouraging of moderate volume in prayer. In Talmud Yerushalmi Berakhot 4:1, we hear that R. Abba b. Zavda used to pray audibly (perhaps even loudly). R. Yonah used to pray quietly in the synagogue (presumably so as not to disturb others), but at home, he would pray loudly enough that the members of his house learned the words of the Amidah from him. R. Mana reports that the members of his father's househould similarly learned prayers from his audible prayer. In Yerushalmi Berakhot 2:4, a baraita teaches that one who prays without hearing the prayer has still fulfilled one's obligation. This formulation suggests that, in fact, it is ideal to pray loudly enough in order to hear one's own prayer and perhaps the only problem is highly excessive volume. On Talmud Bavli Berakhot 24b, R. Huna limits the ban on audible or loud prayer to cases where a person can maintain proper focus. One who cannot focus with a whispered prayer alone is allowed to pray audibly. [A later gloss adds that this is only permissible in private; in public, one must avoid disturbing others. See more below.] Indeed, a tradition of praying the Amidah loudly continued into the middle ages, at least in certain times and places. Piskei Tosafot Rosh Hashanah #72 reports that on Rosh Hashanah and Yom Kippur, it was common practice to pray the private Amidah aloud, despite the fact that this would disturb others.
In truth, this early data seems to reflect an unresolved tension regarding the best way to pray. A truly silent prayer emphasizes God's ability to hear even the most private requests and avoids all pretensions of excessively outwardly directed piety. On the other hand, the passion of prayer, in addition to some of the educational and contemplative benefits of saying the words aloud argue for more space for increased volume. Later sources try to take sides in this debate and to offer some syntheses that can preserve all the values involved.
In Hilkhot Tefillah 5:1, Rambam describes השויית קול--moderate volume--as one of the requirements for praying the Amidah. In Hilkhot Tefillah 5:9, he says that one should make sure to hear one's own prayer, but should only be louder if such volume is indispensable to one's ability to focus and one is in private. Rashba states that one should hear one's own prayer but make sure that others do not hear it. He follows R. Huna in making an allowance for those unable to focus and follows the Yerushalmi traditions by reading them as allowing prayer done aloud with the intention of educating the members of one's household. [The plain sense of the Yerushalmi is that this was a by-product of a loud prayer, not necessarily that it was the intention of the entire exercise.] Tur and Shulhan Arukh 101:2 follow this basic compromise position, though R. Yosef Karo, in Bedek Habayit, later becomes concerned for the Zohar and suggests trying to keep prayer entirely silent. Nonetheless, Magen Avraham and Gra reject the notion that the Zohar should even be read this way and insist that one should indeed pray loud enough so that one can hear oneself. Birkei Yosef and Hayyei Adam affirm the other boundary: one's neighbor should not be able to hear one's prayer.
Regarding the practice of praying loudly on Rosh Hashanah and Yom Kippur, Mordechai Yoma #725 justifies it not only for educational purposes, but also explains that concerns about distracting others are moot when everyone has a printed text in front of them, as was the case on Rosh Hashanah and Yom Kippur, when each individual had a Mahzor. Mahari Weil #191, however, seems to oppose audible prayer even on Yom Kippur. Shulhan Arukh seems to tolerate this High Holiday practice and Rema confirms the practice has continued to his day, but accomodates Mahari Weil by insisting that people still not be excessively loud. Magen Avraham nonetheless encourages quiet prayer even on these days unless it is essential for one's personal concentration.
It seems the best guidance for the Amidah is thus to say it loud enough to hear oneself pray but not loud enough so others will hear. There might be appropriate leeway in contexts where eveyone has a siddur and a person desperately needs a bit more volume to focus, but that leeway must be exercised very judiciously, particularly in public prayer spaces.
What about the other parts of davening? The above sources seem mainly to refer to the Amidah. Is there a premium on praying (at least somewhat) quietly for the other parts of the service?
There are indeed a number of voices that emphasize that the volume dynamics of prayer ought to be the same for all parts of the service. R. Yitzhak Luria (clearly influenced by the Zohar) was careful to have all parts of his prayer, even Pesukei DeZimra, be done in virtual silence. Mishnah Berurah 101:7 endorses this approach. This argues for insisting on a degree of quiet throughout the entire service.
But another line of thought persisted throughout the ages. Ramban, in his commentary on Shemot 13:16, offers the following dramatic explanation of public acts of prayer:
רמב"ן שמות פרק יג:טז
וכוונת רוממות הקול בתפלות וכוונת בתי הכנסיות וזכות תפלת הרבים, זהו שיהיה לבני אדם מקום יתקבצו ויודו לאל שבראם והמציאם ויפרסמו זה ויאמרו לפניו בריותיך אנחנו, וזו כוונתם במה שאמרו ז"ל (ירושלמי תענית פ"ב ה"א) ויקראו אל אלהים בחזקה (יונה ג ח), מכאן אתה למד שתפלה צריכה קול, חציפא נצח לבישה
Ramban Shemot 13:16
The point of raising our voices in our prayers and the point of synagogues and the merit of the prayer of the community is that people will have a place to gather where they can acknowledge the God who created and made them, so that they can publicize this and say in the Divine Presence: "We are your creations!" This is the meaning of the rabbinic statement: "'They called out to God with force'--From here you learn that prayer requires volume; one with nerve can defeat evil."
We do not have an exact full parallel to the source Ramban quotes at the end with its demand for prayer with volume, but the thrust here is clear: there is a power to words being shouted out loudly, just as attendees at a political rally makes their full passion an conviction known with a rallying cry, not just by waving signs. Indeed, this mode of more audible prayer, even when rejected for the Amidah, lived on in many Jewish communities and lives on until the present day. R. Yom Tov Lippman Heller (in Malbushei Yom Tov on Levush OH 101) reports that it was common practice for people in his community to shout out the non-Amidah parts of the service so loudly that the Christians often mocked the Jews for this practice. He cites the Ramban as support for what otherwise might be a surprising practice. Karliner Hasidim until today are renowned for the screaming that goes on in their services. Arokh Hashulhan OH 101:8 also defends this approach and even sees to prefer fervent prayer with volume as stirring up greater passion and intention. R. Amram Bloom (Hungary, 19th c.), in Responsa Beit She’arim OH #40 and R. Menashe Klein (Hungary/United States, 20th-21st c.), in notes on the above responsum, both emphatically endorse praying the other parts of the service out loud.
And while concerns about distracting others might be relevant, they might be mitigated by some of the above factors, particularly given the ubiquity of prayerbooks in contemporary synagogues.
That said, we have already seen serious concerns about being overly disruptive in public spaces, and this extends beyond any local halakhot regarding volume and prayer. In fact, there seems to be a general notion that a person should not behave in a distracting fashion in the context of public prayer. Even if one has a particularly effective spiritual practice one uses in private, one should not import it to the public space if it will not blend in. This is first illustrated by R. Akiva's behavior as described by R. Yehudah in Tosefta Berakhot 3:5:
אמ' ר' יהודה כשהיה ר' עקיבא מתפלל עם הצבור היה מקצר בפני כולם וכשהוא מתפלל בינו לבין עצמו אדם מניחו בזוית זו ומוצאו בזוית אחרת מפני הכרעות והשתחואות
Said R. Yehudah: When R. Akiva used to pray with the community, he would finish before everyone else. When he prayed alone, a person could leave him in one corner and find him in another corner, on account of the many bows and prostrations that he would do.
In the 20th century, R. Moshe Feinstein expanded on this model in Iggerot Moshe OH V:38:6, arguing that one should not do anything strange in the context of synagogue prayer.
To the extent that one prays in a fashion that is so loud and so out of context that it distracts others, it would fall under this rubric of concern. A person always has to be sensitive to their environment. While it feels wrong to silence someone who is passionately engaged in prayer, it is also reasonable to demand that one or two individuals not be total outliers to the aesthetic that is otherwise present in a space. If a prayer space is generally more quiet and contemplative, this person should moderate themselves to blend in more and perhaps find other outlets for more fully expressing their passions through volume.