@Bluddy’s is a fine list! Just to add some of my own notes:
Jamestown is amazing co-op and one of my all-time faves if you’ve got the right friend(s) for it. Jamestown+ is on its way.
Assault Android Cactus is fantastic but I’m not convinced there’s much co-op going on in it! Some amazing levels, the characters are terrific and it just feels so smooth and slick.
Dungeon Defenders is an old favourite and I forget it has split-screen. Tower defence games are great co-op because there’s lots of chatter about where best to place stuff between waves. There’s the original, but ‘Awakened’ is out next year.
Portal 2 is also one of my favourite co-op experiences. It’s just brilliant. I didn’t play split-screen but I can’t imagine it being anything short of amazing still locally.
I can also highly recommend Lovers in a Dangerous Spacetime. Requires proper co-operation and communication but a ton of fun with some interesting unlocks and upgrade decisions.
Rocket League is a classic.
Renegade Ops, Monaco, Rayman Legends (although I prefer the structure/loop of Origins more), Human Fall Flat and Overcooked/Overcooked 2 are all excellent. Make sure you’ve got a strong bond with your friends for Overcooked; it will be tested. In a similar vein: Catastronauts is worth a look if you’re into sci-fi.
I’d like to add Magicka 2 for some slapstick wizarding and comical friendly fire. It has one of my favourite magic systems. So clever and creative.
X-Morph: Defense reminds me of Renegade Ops with its arcade action and tongue-in-cheek premise and presentation but the actual gameplay is a slick mash-up of twin-stick shooting and tower defence. Big bosses, destructible terrain, pathing and you get to be the baddies.
Crawl is worth a look if you fancy a dash of competitive play in your co-op dungeon crawling (or vice versa). You get to be the baddies as well as the goodies here.
Oh! FORCED is an… action RPG sort of co-op puzzler. Very clever and requires good co-ordination. Lara Croft and The Guardian of Light is very similar and also great.
There’s Hacktag, a co-op isometric stealth and hacking game. One player is the hacker, the other the operative.
Joggernauts is a co-op runner game where you alternate positions to overcome obstacles. Very cool idea and very tricky.
Mugsters if you fancy some physics-y sandbox-y aliens B-movie rescue missions with a friend.
N++ if you and a friend are hardcore platforming perfection nuts. Seriously good stuff but not for the faint of heart (like me)!
I hear great things about Heave Ho, and Totally Reliable Delivery Service is worth keeping an eye on if you’re partial to the loose-y goose-y physics and hijinks of Gang Beasts and Human Fall Flat but set in an open world delivering stuff in various wacky ways.
I’m also looking forward to:
We Are Screwed!
Fling to the Finish
That’ll do! Phew.