Games

A few that I've played:

20XX and 30XX - Megaman-style roguelites. Built with 2-player co-op in mind (either local or online) - when playing local, the screen space is shared but the screen zooms out when players are far apart (similar to Smash), so you still get about the same amount of view distance.
Hypercharge: Unboxed - FPS Horde Shooter/Tower Defence hybrid. Up to 4 players (local or online) have to defend cores from hordes of enemy toys, with the ability to place defences to make the job easier. It's mostly about the FPS shooter aspect over the tower defence stuff, though.
Streets of Rogue - GTA1-style sandbox roguelite. Up to 4 players (local or online). To describe Streets of Rogue is difficult. It's a very silly sandbox game where you're given a few objectives (usually neutralising folks or getting an item from another group of guys), but there's always many ways to go about things. It's also one of the few games where the 32 playable characters actually play wildly different from eachother (except the red/blue gangters but that's also the joke), and multiplayer adds another level on top of that as you can easily help (or hinder) eachother's Main Objective, as each class has their own goals.
Helldivers - Top-down shooter. Up to 4 players (local and online, at the same time) get dropped into a planet to complete various objectives. There's a lot of mechanics in this game which would probably kill other games (like how friendly fire is mandatory, or how reloading your gun discards the rest of your clip), but the devs (who also made Magicka, just fyi) made it work very well. There's a sequel coming later this year, too.
Sonic Robo Blast 2 - 3D Platformer. Up to 2 players local splitscreen, or 32 online can play together to run through co-op Sonic levels. Also the only game in this list that's completely free (by nature of it being a Sonic fangame).
A Hat in Time - 3D Platformer. Up to 2 players local splitscreen, and 50 online can play together in a platformer which I honestly find a lot more fun than Super Mario Odyssey. Also, Player 2 is an equal to the main hero this time instead of a glorified "little brother mode" - heck, Player 2 can actually trigger cutscenes and stuff without Player 1's involvement at all. Great stuff.
N++ - 2D Precision Platformer. If you've heard of Super Meat Boy, the original N is the game that inspired it. N++ is just a perfected version of N, with a brand new co-op mode with its own unique levels, which now requires two players to flawlessly move their way through levels where even a single slip-up means starting the level again (though that's fine - they're at max only 60 seconds long, but there's over a thousand of them)
Mercenary Kings - Metal Slug meets Monster Hunter. Up to 4 players local or online. Rise up the difficulty ranks by completing missions, upgrading the parts on your very modular guns and don't forget to eat before you go on a hunt- er, mission.
Pretty much the entirety of the side-scrolling beat-em-up genre, including Streets of Rage 4 (4 player local), River City Girls 1&2 (2 player local) and Teenage Mutant Ninja Turtles: Shredder's Revenge (6 player local and online, yes 6). This entire genre is based on local co-op, so just pick a game and dive in.