Depends on the architecture. If the game has dedicated servers, then they will built 2 separate pieces: 1. The game they ship to the client and 2. The server software which receives messages from the client and propagates world changes to all clients in the same game session.

If the game is PvP (like most xbox games), the there is only the single game, and a 'host' is chosen to act like the server above. This can lead to problems though like host advantage [1], so the networking logic has to be carefully designed in order to mitigate this effect.

[1] http://gamedev.stackexchange.com/questions/73302/is-host-adv...