I will sheepishly admit that as I dug around to try and figure out what is happening, the help information (scroll to the right on your settings page) states that in order to view private photos:
“Important Note about Private Photos:
To access your private photos from Flickr, make sure that your App’s authentication type is set to Web Application and the Callback URL points to https://www.yourwebsite.com/wp-admin/”
If you go to Flickr, this is under “Edit the Authentication Flow” of your App where you generated the API.
Hopefully this will work for you too!