Technically the Form 4 is a tax document, and thus is private. Problem is, the LEO has to sign it, and there is nothing in the law keeping them from retaining a copy as part of signing the form. There's really nothing they can use it for unless they are keeping some kind of local "registry" to know what people have.
If the rule change goes into effect removing the LEO signature, you'll still have to send them a copy to notify them, like you do when you apply for an FFL. Some LEO's will probably keep the info, some will recycle it... who knows.