I HAVE FIGURED OUT THE FIX FOR THIS! ... (Well it worked for me.)
I saw a post from an apple rep telling a iOS 6 user to turn the side switch on to un-mute the sounds. A no-brainier yes, but it made me wonder if the side switch may have been temporarily "confused" after updating to iOS 7 because it added the new settings screen that swipes up from the bottom of the screen. I was using my side-switch to lock the screen rotation and it was in the "off" (down) position (red visible) and none of my notifications were sounding. So I flipped it up and down a couple times which didn't solve the problem. So logically, I proceeded to what inevitably solved the problem...
Go to Settings > General / "Use Side Switch To" and select the opposite of what you are currently using. I had mine on Lock Rotation, so I selected Mute. Then flip your switch up and down slowly a couple of times to see that the confirmation on the screen is correct. You may have to go back to settings again and select the other choice if you originally had Mute selected, I don't know because I originally had Lock Rotation in use.
Hopefully, this will fix your notification sounds (and the lock sound when you press the power button). After using this fix, you can set the use of the side switch to your preference and still have all of your notification sounds!