This is what the grandmasters refer to as "the quiet move". A seemingly innocuous looking move which nevertheless improves the position. There is also the possibility your opponent has chosen the opening Guioco Piano which is "The Quiet Game" in Italian.
No Sound On My Opponent’s Moves

@Stevenq- what browser and operating system are you using? Have you tried switching browsers to see if it helps at all?
@jacklyn
Thanks for looking into it. I use chess.com frequently from chrome (mac) and never tried from any other browser.
If it helps with you log or some thing for debug, i can note down the play/game id if it happens again.

I have exactly the same problem (I can hear my own moves, but not my opponent's). I have IE9 on win7. Just lost on time, since I didn't notice that my opponent had moved... :(
just to add to it (in case it helps the developers of the website).. When this happens, i dont hear the regular moves of the opponents. But i can hear the sound for castlings and when the opponent takes away any of my pieces.

I’ve been preoccupied; It’s only been nine months since my inquiry and I haven’t tested other browsers until now. So far Firefox works well in a 10 game tournament. I guess IE has problems. I haven’t tried Chrome, Safari, or Opera yet…

This problem still exists, there is a problem somewhere. I use firefox, all was fine for me up until last week. Has anyone found a reason and solution to this problem
I have the same problem and it doesn't seem to be getting attention from the developers. Back to ICC I guess.

So far, I have not had any problems with Firefox using Windows 7 in Dell’s Inspiron 580 desktop computer with Intel Core i3 Processor. I have had trouble with ie11 and now my default browser Chrome. In the middle of a tournament ready to begin the round and make my opening move using Chrome, I discovered the piece would not move. I switched browsers and the site worked fine. Some have had trouble with Firefox but, so far, it works for me. A technician should resolve the issue for other browsers. It has been only a year and a month since I posted this complaint…
I've used Firefox all along. But good news, I dropped in for a couple of games earlier this week and it worked OK. Hopefully, it continues to.

Your opponent is probably SWITCHING from sound to no sound just to confuse you. When that happens PLEASE TURN OFF ....your....sound so you can confuse him. 9/10 times an opponent gets the message. I dislike stealth players so I play them at their own game, TWO silent players or 2 sounding players............thats fair.
How is it that when I move the pieces make a sound but when my opponent’s piece moves its stealthy silent?
I have the sound checked on. Is the problem a bug?