For a Nostr relay, "private" does not mean other are unaware of it. Rather, it means others cannot use it. So it's more of a personal relay rather than a private relay. Anyone can see that you are reading/writing from your relay, and they can see its onion address. You can turn it off if you like, but there is no reason to. The more it's on, the better it will serve its purpose as a backup of your activity.