Where did the subject of LAN play come from? This thread is simply about playing splitscreen in netgames.
Best answer I can come up with is, if it was possible at any point, it was probably disabled because having two players on one instance of SRB2 playing the same netgame is almost certainly going to be awkward to some extent. How would F12 work if splitscreen worked on netgames, for instance?
I think there actually used to be a "splitscreen" console variable in really old versions of SRB2, but that may have been removed either to stop cheating or because it may have been broken. I don't know the exact reasons really.