My son is almost 13 and he's been heading off by himself (in the park I am in) since he was 11.
Earlier, actually, if you count using the single rider line and meeting me at the exit as "going off alone", lol. In that case, I'd have to say since he turned seven and was old enough by Disney standards.
Then, between the ages of, say 9 and 11, we did lots of....."You can ride these two rides and then meet me in this spot" kind of stuff.
But, since he was about 11 1/2 he's been trusted enough to spend pretty good sized chunks of time alone in the park I was in. For instance, he would head off to do Sorcerers of the Magic Kingdom or explore Tom Sawyer's Island or do Agent P Missions or whatever and we would set a meet up time and place for two or three hours later. He had a watch, a cell phone, and knew every inch of every park by heart. And, he was never so much as a minute late for a meeting, so as he got a little older, being responsible and trust-worthy earned him a little more freedom.
When we were there again a few months ago, in January, I felt comfortable enough (at age 12 1/2) to leave him in a park doing his thing and go meet up with a friend for lunch at a resort or something like that. Never any problems and I had no qualms doing it.
I haven't sent him into a park alone yet, because he knows the rule for that is 14 and we follow the rules. But, when he turns 14, just over a year from now, I'll have no problem letting him hop a bus or monorail or boat and head off to whatever park he wants while I do some shopping or enjoy a few drinks by the pool.
Of course, we do spend the vast majority of our time at Disney enjoying things together, but he has always been a very independent kid who enjoys exploring things on his own, and I like to encourage that independence and confidence. I think it's important to be able to strike out confidently on your own, and to be able to actually ENJOY that time by yourself, and I think Disney is a wonderfully safe environment to practice that in. Believe me....we have plenty of time together. I work from home and he's homeschooled, so we are pretty much together all day every day! So, I don't mind letting him do his own thing for a few hours in the afternoon each day at Disney. Of course, he does know all the important safety things about how to make safe, smart decisions when he's off without me - whether in a Disney park or riding his bike to our neighborhood Walgreens.
If he was a different kind of kid, I would have felt differently. But, he's always been extremely responsible and reliable and a very serious rule-follower. Whatever guidelines or restrictions I gave him, he would abide by, and I knew that. He was also very good about keeping up with his stuff, using good manners, etc. Otherwise, I wouldn't have turned him loose.