It really isn't a simple problem. At a most basic level you could say that the network provides the same data that the MoveEnemies part of the original loop did. So you could simply replace your loop with:
while( user doesn't exit )
check for user input
run AI
send location to server
get locations from server
resolve collisions
draw graphics
play sounds
end while
However you need to take into account latency so you don't really want to pause your main loop with calls to the network. To overcome this it is not unusual to see the networking engine sitting on a second thread, polling for data from the server as quickly as it can and placing the new locations of objects into a shared memory space:
while(connectedToNetwork)
Read player location
Post player location to server
Read enemy locations from server
Post enemy locations into shared memory
Then your main loop would look like:
while( user doesn't exit )
check for user input
run AI
read/write shared memory
resolve collisions
draw graphics
play sounds
end while
The advantage of this method is that your game loop will run as fast as it can, but the information from the server will only be updated when a full post to and from the server has been completed. Of course, you now have issues with sharing objects across threads and the fun with locks etc that comes with it.
On the server side the loop is much the same, there is one connection per player (quite often each player is also on a separate thread so that the latency of one won't affect the others) for each connection it will run a loop like
while (PlayerConnected)
Wait for player to post location
Place new location in shared memory
When the client machine requests the locations of enemies the server reads all the other players locations from the shared block of memory and sends it back.
This is a hugely simplified overview and there are many more tweaks that will improve performance (for instance it may be worth the server sending the enemy positions to the client rather than the client requesting them) and you need to decide where certain logically decisions are made (does the client decide whether he has been shot because he has the most up to date position for himself, or the server to stop cheating)