In practical terms, copyright-free software would be "viral": you would be free to redistribute under the same (lack of) terms that you received the software under. There would be no mechanism for things like tying source to binaries or the anti-Tivoization terms, but those aren't defining aspects of copyleft.

I'm not sure what "share publicly" has to do with anything. GPL doesn't apply until you do share the software, and even then doesn't require you to do anything publicly.