Oh, who will shoe your pretty little foot? I'm never going back on the pretty little girl Who gave her heart to me. But there are exceptions. I don't have any idea where I learned the melody here -- but since it's from my personal tradition, this is the tune I chose to print.
As you can see, I marked two melodic lines in the chorus. By the time Lord Gregory gets back to marry her, she is drowned or otherwise dead. In general I think that the first group is right; somebody wrote most of those old songs. If you are interested in using a song for publication, recording, performance, or other, please contact the publisher of that particular song.
A few scholars, though, believe that the old ballads arose spontaneously and collaboratively; they have no authors.