> All works under copyright protection that are published in the United States are subject to the mandatory deposit provision of the copyright law
You own copyright over works whether you register them or not. To the best of my understanding that means that you are always required to deposit works when you publish them, whether or not you register your copyright. But IANAL, so I could be way off.
EDIT: Here's a real citation[0]
> What is the difference between mandatory deposit and copyright registration?
> ... Optional registration fulfills mandatory deposit requirements.
So you definitely are required to deposit a copy with LoC whenever you publish/distribute in the US
[0] https://www.copyright.gov/help/faq/mandatory_deposit.html