The funny thing is that I am not even asked to "press any key" on start. Instead, after the initial loading screen, I hear a funny sound, twice. It's the exact sound that plays alongside with the "click" sound when you click on "Options" menu and the one that plays when you go back from "Join" menu. I don't know what it could be.
I played around the setting in the x360ce and I tried so many things. I usually do not resort to forums until I am defeated and have nowhere else to turn which seems to be the case now

Edit:
Also, yeah, I have pressed every single button multiple times and I have also tried using only one controller but nothing seems to work. From what I see around the web, seems like awful lot of players are having this problem. Maybe we can expect a patch or something in the near future that should make it playable for us in co-op?