Nir Rosen, writing in the New Yorker has the best piece I've seen on who's behind the insurgency in town. The short version: It's a melange--including locals and a smattering of foreign jihadis, all of whom are nominally guided by local clerics. As for the Fallujah Brigade who are nomnally allied with the U.S., looks like they spend a good deal of time with the clerics mentioned above.
Another point, Rosen's piece seems to get far more details on the make-up of the guerrillas than anything else I've seen--including say, the NYT magazine's recent, lengthy story. The reporter for that piece doesn't speak Arabic and as an obviously western guy had very limited access. Rosen by contrast speaks the language and was able to hide his American identity. (He convinced the insurgents that he was Bosnian.)
That's not to bludgeon the Times. It's partially about my own doubts. I've done the same thing: Gone places where I don't speak the language, don't have in-depth knowledge of the culture, and then written long, seemingly knowing pieces. I think that story I did accurately portrays the situation I covered. But am I 100 percent sure? No. There's always a tension between trying to convey as much information as possible, molding it into some kind of understandable narrative, and just writing just what you know. And it's worth remembering that when you're in a strange place where you don't speak the language, what you know probably isn't all that much.