Hi Oliver,
Both "Save" and "Save as" are saving web maps only to My content which is different for each user and they have nothing to do with editing. It's like you save an mxd in Desktop.
If you edit something in a feature service ArcGIS Online saves it immediately there is no need to click any save buttons. Thus every user can edit it no matter who owns the data.
Try to edit something with user A then close the map and open it with user B, you have to see the changes.
Akos