I used to do this with Quake II on my PC ages ago; surely it can be done using DOOM. Just launch two copies, each in a window. In one of them, set the session to Host the MP game.
With the other, search for MP games on whatever local IP you're using and presto (by default, the address should be something like 127.1.1.0 or 198.162.0.1 etc.)
At any rate, that should get you going.
Now whether or not you can play at an acceptable framerate/detail level - but (and please don't take this the wrong way) Amiga users seem to be willing to put up with much lower framerates *shrug*.
