Yes. The Universal Translator works on brain waves... for reading. It scans the brain for basic universal patterns, cross references them to translation matrixes. But No, it does not project them, and is supposed to produce audio. This is more obvious with later series, Enterprise, and Nu-Trek. From Memory Alpha:
We establish a "telecommunicator" device early in the series, little more complicated than a small transistor radio carried in a pocket. A simple "two-way scrambler", it appears to be converting all spoken language into English. (Roddenberry 11)
During the writing of "The Corbomite Maneuver" (the first regular installment of Star Trek, following the pilot episodes "The Cage" and "Where No Man Has Gone Before"), the universal translator underwent some further development. Jerry Sohl, the writer of "The Corbomite Maneuver", later explained, "We were originally going to have [each crew member] carry a language translator, which would fit on the wrist like a beeper, and no matter what area of the universe they were in, the thoughts that the people were thinking would automatically be translated into English as they spoke. We got rid of that idea, and assumed that everybody did speak English." (The Star Trek Interview Book, pp. 127-128)
Obviously it is not 100% screen accurate, because this is a cinematic convention. It is used to explain away the English we hear. In Nu-Trek, the translator has an audio delay, and the listeners hear both the original and translated versions, like a real life translator.
Keep in mind its easy enough to explain away with science and technology. Directional audio can be produce with speaker arrays. You can cancel out audio by producing audio 180 degrees out of phase. Assuming a few hundred years of technological advancement, noise cancelation could be real time.
The translator can "detect" when someone doesn't want to be translated.
Lastly, the problem of lip syncing and visual translations. It is never addressed directly in or out of universe. Since Star Trek is Science Fiction, and entertainment, the realism is ignored to make it easier on the audience, and hand waved as "technology got better".
Keep in mind, Star Trek is a universe where things like subspace allow real time communication with no lag over interstellar distances. We can't even talk to the moon without a delay (2.6 seconds round trip). Hell, we still have lag with Earth bound communication.