The Shulchan Aruch states that on RH and YK the custom is to daven (meaning Shemoneh Esrai) in a loud voice (to have more kavana). The MB adds: “but not too loud. And we are not afraid (with the ‘normal’ loud) that other mispallelim will become confused, because most people have machzorim.”
But the Matteh Efrayim states that many Gedolim were against the minhag of davening (Shemoneh Esrai) loudly, and it is better to daven in a whisper (loud enough to just be able to hear oneself), similar to the rest of the year. Perhaps a bit of a louder whisper for more kavanah, but not enough that your neighbor can hear you. The Shelah writes similarly.
Common custom seems to be like the Matteh Efrayim, but one should not protest someone who davens loudly. If it is really disturbing (not just that you are not used to someone doing that, so it sounds funny to you), talk to them after davening.