Not a bad idea.
The easiest solution (and the cheapest one) is probably to have a PC do all the networking. Connect the Amiga to your PC, initiate the connection using a homebrew program, and start playing.
Amiga #1 <-> PC <-> {internet} <-> PC <-> Amiga #2
Reading and writing to the serial port is no biggie, and keeping a TCP connection (or maybe just settling with UDP packets) is pretty trivial too.
Ofcourse, you have to take account for network lag. I have no idea how it will behave in real life.
