You own the copyright when you have it written down in a "fixed form". That includes any manner of recording, music / tab notation, lead sheet, lyric on back of cigarette packet, etc, etc. Ownership of copyright in law requires nothing more.
Proof of ownership of copyright in the event of dispute can be another matter entirely which is why some people go to some lengths to "register" copyright with one organisation, or the other. However, this is only really useful when the the copyright holders are already established and well known, and whose work is at real risk of plagarism.
(The above is written with knowledge of UK legislation. I apologise if it's materially different in the US.)
Anyway, for you and me, we should desire that our work be stolen and a fortune made by the thief in order that, at some chosen moment, when the proceeds are maxing out, we can sue his ass.