Does this game have real online co-op?

[Cherrynobyl]

I remember when the feature was first announced it was just people being able to observe one another without actually contributing to the game.

[YoshiRulz]

There are 2 online modes:
Vanessa's Curse is strictly competitive (and disallows split-screen). It's a take on an "infection" gamemode: A few players are pulled aside at the start to be taggers, while everyone else tries to collect 'crowns' before they get tagged, at which point they become taggers too.
The regular Online Party is the one you're thinking of, where it's the same singleplayer content (or workshop content) with certain objects sync'd between clients. Keys, items which go into your inventory, and other "quest progress" is notably not sync'd. I'd say this gamemode is de jure competitive, since it's set up to have you race other players who enter an act with you from the lobby (including your split-screen partner), but in my experience, people don't take the scoring system too seriously and just hang out.
You can make it more co-operative by using a unique lobby code for you and your friends, and by installing the Online Party Plus mod, which makes it easier to stick together during gameplay.

[Malodorous Flex]

The short answer is no, and that's really my least favorite part of the game. It's great overall, and really fun! But the actual co-op is offline only.