The “publication” status of your module has no bearing on your ability to play online, real-time games with it. You can certainly play this way with a module you only distribute privately. Other people will see your game listed as in progress on the server, but without the module file itself they’ll have no way to spectate/join.
You can also play asynchronous (often called play-by-email or PBEM) games, which never involve the game server at all.