I was a bit confused when you said local co-op. It implies 2 computers on the same local network(connected to the same router, no internet connection required). I think the term you're looking for is split screen gaming.
When I looked at pictures of Spelunky games like Clonk and Liero popped into mind, I think there are options for split screen multiplayer in those games, they're old school PvP games.
lan, local area network. I guess people see it differently.