As I answered via email:
Do you have Steam cloud support enabled on the desktop and laptop?
Check that cloud support is enabled globally (in the client click on the Steam menu and select Settings, and then Cloud) and for the 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; the default should be to match the global setting).
[leaving this for anyone who finds this topic in a search] For the GOG version, in Galaxy make sure you have the latest update available, 2.0.119.430 (A), then click the More button on the D:OS EE page and select Settings. For the global setting, open the GOG.com menu and select Settings, then select Features and make sure the Cloud saves option is checked.
Do you have more than one profile on the laptop? Make sure you are checking the profile that you were using with the desktop.
Did Steam synce when you exited the game on the desktop? In general it should, but sometimes does not, until the next time the game is started.
If you start the game on the desktop and open the load or save window, are the saves listed as being synced online? Check the tooltips; there are different icons for saves that have been synced and new saves that have not.
