Technically, regardless of singleplayer, multiplayer, splitscreen, or dedicated. You still are going to have a server in one way or another. When you’re playing a Singleplayer game your computer acts as a server of information, when you play splitscreen you’re still the server. It’s just a “local” server and you don’t need to worry about information being sent over the internet.
As far as what you’re trying to do, without any indication of what you’re already doing it’s hard to give you a better idea,. There are multiple ways to approach this. It’d be best to only let Player 1 control the menu, and when he selects his side, then the game takes Player 2 and assigns him to the opposing side. But again, without knowing how you’re trying to do any of this I can’t really say.