Hi Carl
So when you save a copy for editing in ArcMap for your hosted feature service, it creates a file geodatabase locally. For me it's in this location:
C:\Users\<user>\Documents\ArcGIS\FeatureServiceLocalEdits
When you initially save a copy for editing there must be a "link" that is established between AGOL and the first edit session in ArcMap to retain the domains, and once you exit ArcMap this "link" is gone. That's my guess anyways!
So essentially each time you open ArcMap to edit your hosted feature services you will need to save a copy for editing.
It's good to know the location of were AGOL is saving these "temporary" file geodatabases, because you'll want to clean this folder out periodically. And I don't think you can resuse them to sync back to AGOL!
It is a clunky synchronization process, that is good for only small editing sessions. If you are doing a lot of editing on a hosted feature service be aware that the default max record return is only 1,000 records, meaning when you save a copy for editing in ArcMap it only pushes down that many records. You can go into AGOL and increase this max record count number, but then you will be sacrificing speed in your web maps. All this to say, I highly recommend doing your edits in ArcGIS Pro, because you do not have to save a copy for editing and you can see ALL of your data. I'm not a fan of ArcGIS Pro, but that's how they have designed it!
Hope this information helps!