Suppose I'm using and client side prediction, wouldn't prediction cause anomalies on other clients? For example, waking monsters. Prediction on one client could cause monsters to exhibit different behaviors than what might have happened.

you can not predict everything.. prediction is very usefull for movement for example. if we know that a players position was X and hes moving at velocy Y then we know he will be at Z in time T.

for something like an RPG where a monster could be sleeping, and stepping near him might wake him, simply just have the server dedect when a player is in certain range of a monster, and then send a "this monster has woken up and is now attacking this person" to all players who should know about it.

Yes, thanks. I should have known that. Vital changes in the world should be handled by the server. As long as the client isn't laggy/doesn't suck, there shouldn't be too much of a problem.