Vermintide 2.

Deep Rock Galactic.

FarCry (I think all of them? It’s been forever since I played 1-3 tho; the newest ones all have co-op)

I also reccomend deep rock galactic, the devs are very involved in the community. The community is (mostly) amazing. There isn’t much story and it is completely hidden in the in game encyclopedia and the wiki. No drm (apart from steam) and lots of content and depth that keeps you hooked once you make it to your first promotion.

Halo is my top pick, the first three have great stories and couch co-op if you need that, and if you get the master chief collection (halo mcc), the multiplayer is great and you can find tons of forge mode maps created by people, from race modes to mini golf haha

Gears of war is also fantastic, epic story, fun gameplay, and who doesn’t love a chainsaw bayonet lol

Deep rock galactic is another really fun one, I can’t say there’s too deep of a story too it, but it’s endlessly repayable with all the combinations of weapons and procedurally generated missions.

Helldivers 2 is fantastic fun and I still play it weekly, even though I’m max level with everything unlocked.

Halo 1 and 2

If you know how to sail the high seas and emulate things, Halo coop is easily my fave.

• Shadow Warrior 2 (avalaible on GOG)
• Half-Life 2 has a mod for campaign coop (Synergy if I recall correctly)
• Lots of zombie games : Left4dead 2, Killing Floor 2, …
• Borderlands

First Person but not totally shooter :

• Dying Light / Dead Island
• Vermintide / Darktide

Remnant 2, it’s third person but it’s one of my favorite games ever.

What do you mean by DRM here? Are you saying you need to be able to pirate it?

When I was a kid I had a blast playing Time Splitters: Future Perfect on my friend’s PS2. Such a fun game, with a great co-op story mode.

Seconding Dying Light 1 and Borderlands 2. Excellent games! If you’re open to 3rd person, Helldivers 2 and The Division 2. No idea about DRM, though.