private.coffee is the address of a server belonging to a privacy focused NGO, who offer all those services, including RedLib, for the general public, regardless of where they come from. They are not affiliated with the fediverse or anyone else.

There are many servers hosting RedLib, all offering the same functionality, to help to preserve the privacy of users worldwide. RedLib itself comes with a variant of the GPL, a licence which gives everyone the right to host a RedLib-service; if you wanted, you could run it on your own computer, or you could rent a server and host it there, offering the same service for others.

This is similar to lemmy - you can rent a server, a domain (like private.coffee or lemmy.world), and host a lemmy instance for yourself and whoever you wish to share the service with. Your Lemmy instance would then talk to other lemmy servers worldwide and share posts and comments with them - this exchange between different Lemmy Instances is called "federation", and it means that no single person can decide how lemmy is run, because everyone can only control their own server. Lemmy also federates with other services, like Mastodon or Misskey, since all those services talk the same language: ActivityPub.

I hope i could be of assistance; if you have any more questions, feel free to ask!

Have a nice day!