If I have 3 drowned prophets out with the "your opponent must discard 2 cards if you win initiative" text on them, and I win initiative, do all 3 instances of this effect stack? i.e. do I make everyone else discard 6 cards instead of just 2?

I assume you mean Drowned Fanatic, not Drowned Prophet?

Note that the discard on Drowned Fanatic is a Response. You have to choose to individually trigger each Response, resolving it completely before triggering the next. So you trigger copy #1 and discard 2 cards. Then you trigger copy #2 and discard 2 more cards. And so forth. So technically, the effects do not "stack" by combining into a single 1x6 effect, but you get the same result by going through the effect 3x2 individual times. There is nothing stopping you from triggering each Response to get your 3x2, though.

(Same thing would be true if they were passive, by the way. Each individual passive would activate individually. So you'd get a 3x2 resolution instead of a "stacked" 1x6, but you would end up discarding 6 cards in total.)