There are two matters to discuss here: whether the davener may act in this manner and what you should do about it. While the former question is straightforward, the latter is more difficult.
The gemara[1] deduces from the prayer of the biblical Chana[2] that tefilla (referring at least mainly to Shemoneh Esrei) should not simply be contemplated in one’s mind. Rather, the words should be formed by the movement of the lips. At the same time, the words should be articulated in a manner that is not audible. The gemara[3] goes so far as to describe one who recites tefilla audibly as one who does not believe sufficiently in HaShem, as he implies that He will not hear the prayer otherwise; one who raises his voice in prayer is further described as a false prophet.
The question of whether it is proper for one to pronounce the words of tefilla loudly enough to hear them himself is subject to an ongoing machloket since Talmudic times.[4] While kabbalistic sources oppose being able to hear the words at all, standard halachic opinions recommend that one be able to barely hear them himself.[5]
The gemara[6] assumes that the main problem with saying tefilla audibly is contextual; specifically, one may do so if it is necessary for his kavana (concentration). The Taz[7] posits that this is justified even to improve one’s otherwise reasonable kavana. (The Bi’ur Halacha[8] objects, out of concern for the kabbalistic opinions that audibility disqualifies the tefilla.) However, the gemara[9] states that even one who must recite tefilla aloud for the sake of his kavana may not do so in a communal setting, because doing so disturbs others. Indeed, common experience indicates that davening out loud when others are quiet is disturbing to others. Although we waive the rule that one must pray quietly for the needs of one’s personal kavana, we do not bend the rule for him if it impacts others negatively.
It is difficult to apply these rules as an exact science. Most people cannot daven loudly enough to hear themselves and still be certain that those nearby do not hear him at all. It is also difficult to know when a quiet undertone will be disturbing to others. Although the words themselves may not be audible, a low “hum” might still be annoying. On the other hand, if a neighbor is overly sensitive, then it is arguably his problem; perhaps he should learn to get used to it or find a different place to sit.
These are issues that are difficult to quantify, and their practical application is often the type of thing that distinguishes average people from those with sterling middot. In other words, one should optimally make sure that he is not disturbing others. On the other hand, the person being disturbed should optimally try to be understanding and give his fellow davener leeway in borderline cases.
I often recall an approach to competing legitimate concerns of different people that I heard from my rosh yeshiva decades ago. The Torah instructs that it is forbidden for a lender to harass a borrower who is having difficulty paying back his loan,[10] but a different pasuk has sharp words for a borrower who does not pay back.[11] Although these principles can sometimes be on a collision course, both are true values. The intention of these p’sukim is that the borrower should focus on his obligation to pay and the lender should focus on his patience in getting paid. The problem arises, as can be anticipated by human nature, when the borrower is too fond of the pasuk intended for the lender and the lender is too fond of the pasuk meant for the borrower.…
Even if one may request that the audible davener conform to the Halacha, it is not always clear whether and how he should do so. The Chazon Ish[12] made famous the opinion that in our days, we do not know how to give rebuke. I would add that our generation is particularly unsuccessful at accepting even minor correction (not that it was ever easy). You therefore must think about (and/or discuss with someone who knows your neighbor better than you do) how he will react, externally and even internally. This may depend to some degree on how he is told. Should the message be conveyed directly or in a carefully worded (anonymous?) letter? You may consider asking the shul rabbi to speak to him or perhaps to discuss the halachot with the community. What works best in one case may fail miserably in another. Depending on the prospects, it might be better to put up with the situation or to move your seat discreetly rather than to say something.
[This response is addressed to the person who asked and to readers who are similarly bothered by audible daveners. For those who practice (possibly) audible davening, we suggest you discuss the matter with someone you respect and avoid disturbing others and/or having someone complain to you.]