I've used keyboards with MX Red/Clear switches and thick PBT/POM keycaps for months/years in classes without any complaints; I never bottom out hard, thus dampeners don't really make any noticeable difference and I dislike the feel (shorter travel).
Dampened (Matias) Alps are even quieter. So are some better scissor switches, especially if you put a rubber skin over them (like these for thinkpads, macbooks or typematrix). Actually, nearly any rubber dome will sound similar, provided sliders aren't mounted in a creaky plastic case, but rather on a metal plate or something—sort of like Topre, although it doesn't have to be Topre.
Everything I've mentioned will make some sound, but a not the horrible creaking noise when someone smashes a cheap rubber dome over membrane, rather a distinct clacky sound.
More importantly though, put your keyboard (whatever it is) on a soft surface, like a thick cloth mousepad, a towel, your lap or one of those faux leather deskpads. It's amazing how much noise it absorbs.