You know, I think the way people respond to the bans actually makes them more dangerous than they were before.
From completely non-scientific personal observation: Before the ban in my state, I could pick out the distracted drivers easily because they had glowing objects in their hands on top of their steering wheels. They also had less distance to go to look forward then back at their screen, which made them relatively safer. Once it was banned, drivers became instantly better for roughly 3-4 months. Once they realized enforcement was lax and texting loopholes, the driving rapidly became distinctly worse than it was before the ban. Traffic is more severe, people failing to notice traffic lights became a common occurance, etc. The texters simply moved to texting down low, the headset crowd continued using their headsets, and others went to using speakerphone (which is arguably worse than holding a phone to your head because it's more mentally distracting. The net effect was worse and more dangerous traffic, though I would believe that accident rates are no higher because some fraction of drivers are more aware than those that have become even less aware.
Mind you, according to this study, I'm an exceptionally dangerous and reckless driver that should get into accidents at an above-average rate. Except, of course, that I don't.
Oddly enough, the best traffic I've ever encountered was in Europe, followed by remote rural driving. The latter is simply due to low numbers of cars, which agrees with Smeed's Law. The former, I believe, is because there's more barrier to entry for driving combined with excellent public transit and no public transit stigma - average middle-class people might live their entire lives without ever learning to drive, which reduces the number of cars on the road and allows realistic self-evaluation. This is what is missing from countries like the US.