I think it would depend upon how mature your kids are. My mom still doesn't completely trust me to go off by myself and I'm 18. If I do she calls me on my cell about once every half hour. But seriously, I think the first time I got to go off by myself I was 14. I was allowed to walk from the hotel (Yacht Club) to Epcot and do whatever I wanted for an hour until I had to meet up with my aunt. Whoop-de-doo a whole hour by myself, but at that time, I'd take what I could get. Keep in mind that I was pretty much raised in Disney World and we go every year so I know my way around the parks like the back of my hand, how things work (i.e. what to do/what not to do), and where to go for help if I should need it.
If you trust your kids enough to let them go off by themselves, lay out some strict rules and if they have a cell, call them every now and then to check up on them. They may not like being checked in on and having rules (I know I didn't and still don't) but you can't trust everyone out there, even at Disney World and I've kind of learned to appreciate it. Also show them where the lost person/first aid/information office(s) are located.
I know I kind of strayed from the what age thing, but it really depends on you. I mean no one knows there kids better than their mom..