It turns out I need to keep a closer eye on things before commenting. The ability to transfer ownership of a GM-controlled PC is not yet exposed in the UI. My development focus is the backend, and that capability is in place there, but it has not yet been surfaced through the UI. So a GM-controlled/owned PC in a campaign with content-sharing can only have its controlled transferred at the moment. Apologies for the confusion.
Each of the options is available through the vertical ellipsis menu to the right of the PC on the "PCs and Players" tab. However, only one of the options will be found right now. The other will be appearing in the not-too-distant future.