I guess it depends on the MMO, both implementations can be made. Actually, it is a little bit of both.
I'd be tempted to say that generally the server is to handle everything world related (helps preventing some cheating).
The client is merely rendering the world/view of the player accordingly to the informations sent back from the server.
Also, there's some prediction made in the client itself, which is then "tempered" by the server's authority.
Perhaps this talk would be of interest about this question:
[tube]tKSYJYV_RGs[/tube]