John said that it was a Paul's song and he helped a little:
"Paul wrote the main structure of this when he was sixteen, or even earlier. I think I had something to do with the middle." (1972)
"'Love Me Do' is Paul's song. He had the song around in Hamburg even, way, way before we were songwriters." (1980)
By the way, tomorrow is October 5, 2012: 50 years after the release of the "Love Me Do" single.