wallet – Can everyone see “message signatures” I create & verify?

I need to sign and verify messages on a couple of addresses, and I need to write my full name and address as message. Is there any concern that anyone can read the message(name and address) other than the individual I am revealing it to?

I’m not sure how signing and verifying works other than that it shows proof you own the keys tied to that address your signing for. But is the message when signing and verifying sent out on the blockchain out in the public so everyone can read and see the message?

Any safety/privacy concerns here? Can everyone read my message online afterwards?