I certainly am no stranger to handguns. I love all guns and have quite a few. I used to carry all the time in the backcountry, and stopped after I discovered that in 20 years of packing that iron, it had never been needed for protection.
I lived in Cochise County for a while and did lots of hikes in the Chiricahuas. As others above have noted, I would be apprehensive down by the border there today without serious firepower, mostly around the back roads and trailheads where cars are, not so much at 9k in the backcountry.
If you are considering carrying, I have three suggestions. 1. Carry enough gun to do the job. If you run into a really mad bear or a really bad person (guess which is worse), they will only be worse company if they are wounded but not put down. Our special ops folks in Iraq no longer carry a 9mm, they've gone back to my favorite, the 45 auto. 45acp, 40 S&w, and .357 mag are all pretty effective stoppers, .38 with +P overloaded ammo is acceptable, 9mm I just don't have much faith in, especially on thick hide or heavily clothed targets. Don't shoot bears in the head, the skull is amazingly thick and even heavy rounds can glance off. Aim for Chest/body mass. Same is true for the two legged critter, head is small target. If really nervous, remember that any center fire rifle is more powerful than a hand gun at close range. If it's still moving, keep shooting. Reload, reload, reload.
2. Know how to use it well, before the crisis. Remember how automatically you acted the last time you had a close call when driving. You won't have time to think about how the mechanism works when the need arises, you'll rely on practised response. This means practice....
3. Be emotionally and psychologically prepared to use it in appropriate circumstances. Really bad folks are the best psychologists in the world. If you have it, but are afraid or unwilling to use it, they will not only know it, they will smell it. There is then a very good chance that it will be taken from you. This happens, and is probably worse than not having the protection in the first place.