It should, unless you get disconnected (in Live). It should also notify you on Daily unless you have e-mail alerts turned off and in your alerts when you log in. Also, in Daily, if you have the tab open on the game and they make a move it should show you.
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I seemingly have the same issue in daily games. Frequently, if I have a daily game open where it appears it is my opponent's turn, and then go back to the daily page, then I will get an indication that my opponent has moved. It behaves as if there's some sort of lag. I particularly note that this is when I am "live" playing a daily game with my opponent. It might have something to do with my settings because my opponent's move speed suggests they don't suffer the same lag, but if it does I don't know how to correct them.
why doesn't chess.com let me know when my opponent has moved?