Jace's Erasure with Prosperity
The order of events in this case is as follows:


Prosperity resolves
Resolve the text on the card in its entirety, so each player draws X cards.
Jace's Erasure's ability is triggered once for each card drawn, but we wait until the resolution of Prosperity is complete before putting the triggers on the stack.
The triggered abilities of Jace's Erasure resolve one at a time, and you mill cards from your opponent's library one at a time as they do.


Summary:
You both draw, then you mill the cards from your opponent's library. Changing whoever cast the spell does not affect this outcome.