It is extremely simple to export a single player game, but I have not been able to find a way to create/export a multiplayer game from the BGE.
Is this theoretically possible without using a third party application?
Short answer: No, at least not without a lot of effort.
The engine has no baked-in network multiplayer. You have to set up the whole thing yourself with Python scripting. It's all quite complex.
As far as I know this is the most in-depth explanation of the subject: http://youtu.be/4xZRfzOtxzA
^ Should you decide to build that into a full game, be sure to swap pickle for something like json. Using pickle on untrusted data is a nightmarish fiery hell of insecurity.
If you're familiar with Python and compiling, check out the Python bindings for ENet: https://github.com/aresch/pyenet
At one point someone was working on an addon to ease the burden for common folk, but last I checked it wasn't matured enough for actual use. The project might be dead as well: http://blenderartists.org/forum/showthread.php?242740-Blender-Game-Engine-Multiplayer-Add-on