Multisite privacy/request access

Hello
I’ve got a mutlisite up and running but I can’t seem to find a way to get it to do what I need:
I’m looking to have most of the sites be accessible to the public but have one or two private with some way to request access to them. Has anyone pulled this off or have a clue how to? I’ve searched high and low and through as many plugins as I can count but nothing’s quite filled the requirements.