This is not a bug, but how you implemented the multiplayer concept.
First problem: for the synching, you should add the client input tags, "xpeer" and "ypeer" on event 8 in the game event sheet. (like lookx looky and input tag)
Adding those will fix the host movement on host and peer.
As for peer movement, there is no peer control structure implemented.
Simply copy the control pieces from the host group to the peer group. (event 29 in host group) (or move event 29 to the common group)
And voila ... it will work. (just tested here)