the game lists co-op play but is it single screen 'couch' co-op with multiple joysticks or is it network co-op so I can play it with my children on our LAN?

thank you.

I'm also looking for details on how co-op works.
Specifically are there options for local co-op like LAN and hot seat, or anything like individual hosting that can be used locally without calling in to a remote server?

'Couch' co-op. 2nd Player's character is tied to the save and has their own separate levels and equipment etc.

Personally, I would really like the game to have LAN co-op in the future (There's a big GOG Not Connected status on the game menu), but I've read on Steam from the dev that it's not likely.

Fun game though. Some issues with couch co-op UI and QoL issues, but the amount of content makes up for it.