Q. Why do we prohibit using a microphone on Shabbos, and we permit speaking to a person wearing a hearing aid?

A. Igrois Moishe (O.H. 4: 85) explains that microphones became prohibited by most Poskim because they involve “hashmoas kol” or an activity that is publicized and creates awareness to all that a prohibited melocho is possibly being transgressed, which is not the case with a hearing aid.
Also, he adds, only a small amount of people in need require hearing aids, thus it is a “milsa delo shechiach” or an uncommon occurrence, that our sages usually do not prohibit.
See last question, where we quoted an additional reasoning that in reality the melocho is not done by the hard of hearing, but by the one who addresses him, and he is “eino mechaven” (does it without intention). Igrois Moishe maintains that it is not an unavoidable melocho prohibited also when done without intention, since the hard of hearing may not even be listening.
Horav Shlomo Miller’s Shlit’a opinion is similar.

Rabbi A. Bartfeld as revised by Horav Shlomo Miller Shlit’a