I thought I had figured out a solution, but it was a false positive.
I created a second field on my Image Thing and made it an explicit ‘image’ type field instead of a text field (which is what the external image URL field is).
Still, in either case, when the image is saved to Bubble it get assigned the userID of the logged-out user A.
The initial question remains: how do you control file ownership in cases where Bubble saves files directly to the file manager?