Would have been helpful if you noted what genres you like, so because that was not mentioned am going to avoid anything too retro-
Divinity: Original Sin II is my suggestion, Strategy RPG game which is very fun with another. I have played coop with a partner and can personally recommend it as a fun experience and gives both the freedom to play as they would like. No screen sharing either you can both split up or group together whenever you both feel like it.
Once Baldur's Gate III is released (currently in early access) would be another good suggestion, BG3 is also Larian Studios and features the same kind of coop experience as D:OSII; difference is BG3 has a much bigger budget and will feature a lot more options.
Like play with she quick runs in small games such as .io browser games coz then we do different offline things together ;) We love Fireboy and Watergirl, so you can check this and some together playing games here for example