Asklemmy

Note: unfortunately these are all on PC and don't have couch coop unless specified otherwise. I also don't think they have DRM aside from normal steam stuff (or require an internet connection for actual gameplay purposes), but any steam DRM should be easy to remove.

Warframe: space ninjas: the game (though it does require an online connection since it's basically a coop mmo, and it is third-person as opposed to first-person)

Payday: basically cops vs robbers. You're the robbers.

Killing Floor: Lodesemone! Dosh! Grab it while you can, lads! Money, money, money! Coop zombie survival. I hope you love blood and gore.

Abiotic Factor: what if you were a scientist in the ~~Black Mesa~~ GATE Cascade Research Facility and you had to survive? Can be played single player or multiplayer.

Halo: MCC: Halo already has a reputation for being the ultimate coop shooter. The MCC adds and expands coop for the games; however the PC version lacks split-screen. This can supposedly be added back with a mod, but I haven't tried it so I can't attest to how functional it is.

I've heard Sven Coop is really good if you like Half-Life stuff. It's a coop Half-Life mod.

Risk of Rain 2: technically third-person, but a very good coop shooter.