I found this link to a similar problem.
It's from 2012.
So now, it's 2020, I am using Outlook 365, with an on-premise server, and I still seem to have this issue.
We have a shared mailbox, lets call it HR.
This mailbox is shared between several people. Lets say one is Joe.
Joe logs into his Outlook account. He was setup with Full Access and Send Access in ECP, so technically should be able to do everything in the HR shared mailbox. He wants to move some files around within the HR mailbox. Not out of it to his mailbox, or somewhere else, but just WITHIN it.
However, he finds that for any folder that contains a folder within it, it gives an error message if he tries to move it. Or delete it.
The only solution that I have found is to manually recreate the new folder in the new location. You then have to copy the folders that reside in the original folder location to the new location, and copy any emails over. Sloppy and error prone.
Now I've seen some solutions that say you need to give people Delegate Access, by logging in as the Shared folder, and then right clicking the inbox, and setting Joe as a delegate, but this is Shared folder on Exchange. It doesn't have an account that lets you log on. And Joe already has Full/Send Access on the HR mailbox.
What is the solution here? I see this one, but I don't know how I would grant them rights to "Private items"