Try verifying local files: in the Steam library, right click on Divinity Original Sin Enhanced Edition and select Properties, switch to the Local Files tab and then click on the 'Verify Integrity of Game Cache...' button.
With the GOG version, in the (optional) Galaxy client, select Divinity: Original Sin Enhanced Edition in the left column, then click on the More button, and in the Manage menu select 'Verify / Repair'.
If the client system is seeing quests differently than the host (and the host is accurate), maybe something is being carried over from a different save.
If applicable, on the client system disable Steam cloud support either globally (in the client click on the Steam menu and select Settings, and then Cloud) or just for this game (in the library right click Divinity: Original Sin Enhanced Edition and select Properties, then switch to the Updates tab and check the Steam Cloud section).
Next, try browsing to the '..\Documents\Larian Studios' folder and rename the 'Divinity Original Sin Enhanced Edition' subfolder.
This folder contains the saved games, configuration files and a level cache folder. Deleting or renaming it will get the game to recreate it on startup; playing the game from a different Windows user account would effectively do the same thing. With Steam cloud support enabled, Steam would just download the cloud copy of your existing profile.
After that, start D:OS, create a new profile and then try joining the co-op game. If that fixes the quest problems, exit and move the saves over from the renamed folder into the newly created profile's savegames folder.
If a new profile doesn't make a difference, delete the new My Documents D:OS EE folder and rename the original back again.
If the above doesn't help, try the same on your system: rename the D:OS EE folder, start the game and making a new profile, exit and copy just your latest save to the new savegame_patch folder, and checking it in-game.
