Nostr is at its core a public system, but because it uses encryption, you can absolutely store and send information privately over its relays. The confusing aspect about mute lists is that there are both public and private lists, but they are not supported in the same way by all clients, and, as it turns out, some clients even overwrite private lists without warning you.

If backing up your public and private mute and follow lists to keep them from being unintentionally deleted is a priority for you, you should use #Mutable. I built it to do just this, because it had happened to me one too many times.