No, that's not how it works. At least in the US, an author has an implicit copyright over his or her work even if he or she does not post a copyright notice. The author would have to explicitly forfeit his or her rights to the work in order for it to be in the public domain.
This was actually the case in most of the world when the USA was still using a register. It's relatively recently that the USA has signed up to what the rest of the world has been doing for a while ... just for a change.
Of course there are some places that haven't signed the Berne Convention and don't have a copyright treaty through TRIPS or something similar.
