Use a script to copy over the photos from the public to the private. I assume the software ur using has a database to store all the metadata. U can set up one of ur servers as a master db and mirror that through a variety of tools to the other location. I think u would need to set the master db as the one on ur private machine (as it has all the content) and mirror that to the public. I’m not sure how ur software will handle having metadata for images it doesn’t actually have.

Their might be some way to do bidirectional db sync. Or perhaps run the db on the public and connect ur private to that db. So u would have 2 services running off the same db (and syncing files back to the private as necessary).

I use Immich as my image management and u can point it to any db service u want.