With extreme difficulty.
There are quite a few lists of films that have fallen out of copyright, or were never registered correctly, but even those often contain elements, like the soundtrack, that may continue to hold copyright.
Secondly, what is Public Domain in one nation may well not be in another. I know of several films that are public domain in Australia, that won't reach public domain in America for another fifty-seventy years, and vice versa.
Legally speaking, if you're broadcasting for everyone without checking where they live, such as through the registration form, you can and do run up against people willing to litigate you to death over a film made in the 60s that hasn't seen mass broadcast since.
Finally, finding out who owns copyright of a work can be extremely difficult. A company can go under or get carved up, and various trademarks and copyrights can be passed around - often to different people, and occasionally, the copyright can be lost in that mess or owned by an entity that no longer exists. It's still not public domain - but nobody knows who should be angry at you if you broadcast.