I think the copyright is automatically given to whomever releases the project (completed/in progress, not script) or whomever emails a version of the script to themselves or someone else, because both of those are time-stamped and obviously credited to the creator. Contracts from production clarify ownership further but those don’t have to be registered or a matter of public record. Registering a copyright in a lot of cases is a formality, but fair use wouldn’t be a thing if copyright wasn’t assumed if you literally release the content. You make it, you prove it, you own it, I’m pretty sure.
From this link:
Many artists and writers fret over copyright and intellectual property laws without realizing that a work is essentially copyrighted as soon as it’s created. When a work is created, it doesn’t need to be registered with the U.S. copyright to qualify for copyright protection (though that will strengthen the protection a work can receive). It just has to be “fixed” in a tangible medium of creative expression.
A work becomes fixed under copyright law when it’s put down in a tangible, perceptible form like paper, film, audio tape or email message. Unrecorded speeches, live performances and live TV broadcasts that are not simultaneously recorded wouldn’t be protected because they’re not considered fixed, but a manuscript typed up and sent to an email address or a song recorded electronically would be fixed.
I’ve only registered two scripts with the WGA, both times to submit full length pilots to contests or fellowships, but otherwise I haven’t bothered. I guess in that case, I would register with WGA if I was submitting the script to anyone/anywhere that I didn’t know personally, just to cover my ass.