2D Multiplayer Game Considerations Thread

I am currently about to embark on my next big game project and I find myself stuck deciding how exactly I should approach it. My goal is to make a 2D, arcade style, fighting game that I aim to try to get onto Steam. In the past, I have used libGDX and Java to create cross platform apps and games, but for this game I want to get away from Java, and ideally use C++ or C#.

However, my main issue is this: This game I am creating requires the ability for multiplayer, both as LAN and on a server. I have never really dealt with the multiplayer aspect of games, and I don't really know what engines are good for that kind of thing. Does anyone have any recommendations for a 2D engine that feature good multiplayer support, and can be exported at least to PC and Mac? I also am considering the fact that if I want to make a really solid game, I may have to build my own engine, but I only want to approach that if that is my best option.