I think it really depends upon the child whether they are mature enough to handle the responsibility of going on their own. Ask yourself this. Do you currently allow them to wander the grocery store or department store alone? As you said,
Disneyland is much larger, but if they handle that okay and feel comfortable on their own and get back to you when they are supposed to, perhaps they are ready.
I will be honest I am too tired tonight to read through all of the replies, but I did see someone post about the what ifs, earthquake, accident and the child needed medical attention, etc. I would keep that in mind when making your decision and perhaps only allow short jaunts on their own. Have planned times to meet back and tell them to check the time on their phone often and if they will not be back on time due to lines to let you know.
Make sure they know what to do in the even that you do not respond to your cellphone because you do not hear it or do not feel it if you put it on vibe, believe me, this has happened to me at the parks. Also, tell them what to do if they lost the phone or the phone battery died or just bad reception and could not make a call. I would have them go directly to City Hall in the event that they can't reach you, and if you could not reach them, eventually you would find them there.
After today's tragic events, it does bring to mind that there are crazies out there and I do remember a couple of things that happened at the resorts that were certainly unexpected. Several years ago there was an incident at the Grand Californian Hotel where my husband encountered someone who was naked in the elevator. I had almost had my son run up to the room to get what we had forgotten when my husband said he would go, it was pretty late at night and obviously they were drunk. There was a report of an almost child abduction at World of Disney a few years back as well. I am not a paranoid person but again just consider everything before sending them off on their own.
Do they each have a phone to carry? Just wondering what if they get separated while walking through a crowd or one uses the bathroom and the other comes out and can't find the other for whatever reason. I am a planner like you so I figure you will want to have all your bases covered. Tell them each what to do if they end up losing track of one another.
All of that being said, we have let our son go on his own around the age of 12 (I believe) but he had also been to Disney many many times and knows his way around the park very well. He has always been just fine.
Have a safe and fun trip!