I too had this problem and I'm not sure when it started, however the recent update to 16.34 did not fix this. However, in my case at least, it wasn't a Microsoft problem although the symptoms shown in this post did appear and did so only fairly recently.
The problem occurred irrespective of the application used - e.g. creating a file in the textedit app.
The symptoms I experienced were that I could create a file and update it no problem, but my wife could only read it - an attempt to update it was met with those error messages about sharing violations. Vice-versa applies as well. We were trying this with
a file in a Shared folder - the one OSX creates in the /User/ directory when you have multiple IDs set up. I can't remember if we'd used this directory to share files we'd both want to update in the past or not now unfortunately so I can't tell if it is an
issue that arose with Catalina. Interestingly (or frustratingly), no application (Office or otherwise) indicates that it is opening the file for Read Only, only giving the errors when a save is attempted.
So I can't say this isn't a Microsoft problem as there may be more things going on for some people - others seem to have cleared it up by rolling back to an older version or giving the application full disk access rights through Preferences (this didn't
work for me.) It could be that OSX Catalina is being more strict about application of rights (perhaps fixed old bugs!) coupled with changes to some apps.
What did work for me is to set up a new Access Control List (ACL) and assign it permissions then assign it to the folder and contents. In Preferences, Users and Groups, add a new Group - say, SharedEdit - and add your user(s) to that group (tick the ids).
Then in terminal run this command:
sudo chmod +a
"group:GroupName allow list,add_file,search,add_subdirectory,delete_child,readattr,writeattr,readextattr,writeextattr,readsecurity,file_inherit,directory_inherit"
"/Users/Shared"
GroupName should match the name of the group you just created, e.g. group:SharedEdit
The last part of the command should be a path to the folder that your file(s) are in, e.g. "/Users/Shared"
For some reason I don't fully understand, but is related to the way OSX (Posix) applies permissions, it's not enough to set permissions through the folder's/file's "GetInfo" pane.
Further, if the folder or file is on a drive that is removable, open GetInfo on the drive itself and make sure that "ignore ownership on this volume". It should get set automatically if OSX recognises such a drive but it's worth checking.