The freedom of choosing where to throw down your tent or hammock is part of the fun of hiking the AT. I almost never slept in shelters, and would generally prefer (especially towards the end) not even to pitch tent near them. Often enough you find areas that have become established unofficially through frequent use -- it is not hard to see where tents have been laid down and feet have trampled the earth.
But sometimes you will find yourself watching the sunset and sweating a bit about where you're going to sleep, and in these times wandering off the trail a few tens of yards into the woods to find an acceptable spot happens.
There are really only a few places on the entire AT that you are not allowed to camp outside designated areas. The first and most draconian is in the Smokies. Just follow the rules there. The second was in MD, where they want you to camp at designated sites, but as long as you don't plop down along the river outside of Harper's Ferry, no one will really care. NY was the same way - 'designated sites only', but that's not really how it ends up working. You're not allowed to camp in the alpine zone in the Whites (or in Maine), and this is because the alpine vegetation is so fragile and the environment can be dangerous for ill prepared (or even well prepared) individuals. Within the Whites pretty much anywhere else is okay as long as you're not within a quarter mile of a shelter. And finally, nowhere in Baxter State Park (although I saw plenty of great places off the trail you could probably get away with it... not encouraging it, but I considered the possibility). Maybe there are a few other places people can remind me of.
Anywhere else, that is to say the remaining overwhelming bulk of the trail, is fair game. Even in Shenandoah National Park you're allowed to stealth, as long as it's not within sight of roads or whatever (that is their official policy as I recall, and I had to do stealth there on a couple occasions). They couple it to a charge that you try to practice rigorous LNT. Indeed, in SNP it is quite rare to see established flattened earth where people have camped aside from near shelters, but that doesn't mean you can't wander into the tick ridden undergrowth off the trail and find a flat-enough spot to set up.
Basically, camp where you want to, but be responsible and safe about it, except in the alpine and in GSMNP where you really shouldn't screw around.