My 13 (and a half) year old DD did a local theme park with a friend earlier this year. I was nervous about it, but she kept in touch via cell phone, followed the rules about staying with the friend at all times and all was well.
I think I would be even more inclined to allow this at WDW, because it's a better overall environment than the local theme park, and also one that DD is more familiar with. She brought a friend to WDW in February, but they didn't get to do the parks alone that time, because the friend's mom wasn't comfortable with that (which is perfectly fine, I was just thrilled that DD got to bring a friend.)
It definitely depends on how mature and responsible your child is, and also how comfortable you feel with the situation.