It's not that "you can record only public domain songs", but that public domain songs (and your own originals) are the only ones you can make recordings of, to SELL to people, without having to pay anything.
Current statutory rate is 9.1 cents per recording printed (intended for sale) -- in other words, not as sold, but as printed. So if you print 500 for sale, at that point you owe $45.50 to each copyright holder. A common way to research ownership is to look up the song on harryfox.com. They even facilitate online payment.
It can be hard to determine if an old song is actually P.D. or has a valid copyright claim (for example, many compositions labeled "A.P. Carter" are not actually his compositions, but songs with unknown origins, that he claimed to own -- in keeping with practices of the time). Many songs that are actually P.D. have a variety of people claiming authorship and/or ownership on works they put out. Those claims are rarely contested, and sometimes unnecessarily treated as valid. The law and precedent are a bit fuzzy, and to complicate things, copyright law has recently changed to allow a valid copyright to last longer than it used to. I believe it's now 75 years after the death of the composer.
All of the above muddies the legal landscape and all the while, it's not at all uncommon for people to not follow the law to the letter, with no legal consequences. In general, however, I'm pleased to report that any song fairly recently written, with unambiguous authorship, is normally paid on quite properly, at least for recordings. There is a lot of illegal printing of words in songbooks and published on the internet, but rarely is there legal action.
Practical advice: If you see a song you know is old, that's sometimes referred to as P.D. (look on album covers and in songbooks), and sometimes as being supposedly written by someone, it's very often actually P.D., and VERY unlikely to cause legal action in any case.
But exactly how to find out for sure that a particular song is P.D. is not an easy matter. If enough money is involved, it could easily end up in court.
You can find some more info about copyrights and such in my book How to Make a Band Work. Watch out, though, since the law has changed, and can change some more.