As I understand it, one can often purchas a "compulsory license" to record a song after it has been recorded once. I think (not sure) these are purchased through the Harry Fox Agency. If compulsory licenses are available for a song, then anyone who buys the license can record the song. One then (again, this is not legal advice, I am not your lawyer, don't take my word for any of this. Get a lawyer before you commit your money to anything) must keep accounts of every sale of the record containing the licensed song, and submit monthly reports to the Harry Fox Agency along with mechanical royalty payments for the licensed song. I forget the formula for computing the royalties.

A statement that one need not "get permission" might be considered to be technically correct in the sense that, if compulsory license is available, one can buy a license and therefore need not get explicit permission from the copyright holder. But as Marc suggests, such a statement is extremely misleading. As I understand it, one must have a license, whether compulsory or negotiated.

This is all discussed in Krasilovsky and Shemel's "This Business of Music", 7th Edition (1995). Anyone who wants to enter the music trade probably does well to acquire a copy of this book.