TLDR: Controller-based online co-op PC games that two guys can pick up once in a while after a long day at work and deal with absolutely no bullshit.
Fucking HELLDIVERS, mate.
I hear it has some similarities to the Magika games, so might be worth looking at those as well.
Just saw a stream of some guys playing Red Solstice, looked pretty hot, but more co-op real-time tactics than twin-stick shooter. Have a look and decide for your self though, might be up your alley.
There are also a metric assload of 40K games out now, I'm pretty sure at least a couple of them are real time shooters. Kill Team comes to mind, have not tried it myself.
I hear good things about Vermintide, I guess it is a Left-for-Dead-like, whatever that means? First person class based buddy-shooter thing. Have to stick together and leverage your class synergies to avoid being overwhelmed. Some friends were trying to get me into it, but FPS isn't really my thing.
On a historical note, the Starwars Battlefront games were my absolute favorite crack-a-beer and play with a friend after work games. I'm not sure how they have aged, and I hear the multiplayer servers are down. We used to play split screen on the sofa, and some of them might have private server or local coop options?
Also, I think the SW:BF formula of fighting over bases that act as spawn points until one side controls the map -- I think that has basically been copied to fucking death, so I am sure you can find some other 2-player co-op games that do the same thing.
Another one (and this is a guilty pleasure of mine) is Shadowrun Boston Lockdown, which is co-op turn based tactical shooter. it is a "bad" game by any measurable standard, but somehow it is way more fun than it has any right to be, and I wasted a lot of hours playing it with my buddies. Really had a great time with it, despite it's numerous shortcomings.
I am currently enjoying Satellite Reign, which is a spiritual successor to the original Syndicate games. Dont know how it plays with a controller, I use mouse and keyboard, but try asking on the steam forum? Multiplayer can be a bit laggy, but it is still loads of fun.
TL;DR:
HELLDIVERS
Magicka?
Red Solstice
random 40K franchise crap
Starwars Battlefront
Shadowrun Boston Lockdown
Satellite Reign