Syncing Desktop and documents to iCloud
My understanding is that there is only one desktop synced per iCloud account. I would remove everything from the desktop of an 'about to be connected' mac - so you can manage any clashes. 

Add your Desktop and Documents files to iCloud Drive says 


  You can organize and combine folders from multiple Macs when you turn on Desktop and Documents on those computers too. After you turn on Desktop and Documents on a second Mac, you'll find its files in folders with the same name as your second Mac inside your original Desktop and Documents folders. Then you can combine your files manually, but iCloud doesn’t automatically merge the files in case you want to keep them separate. Any edits you make are seamlessly updated in iCloud Drive. There’s no need to manage your folders and files again on your other devices.