Hunters argue about shot placement all the time. What is best? Behind the shoulder? The neck? On the point of the shoulder? Everyone has an opinion, or at least should, because shot placement is very important in the field. When hunting, you're trying to kill the animal you're shooting at. With self defense, things are a bit different.
If you have to shoot someone in self defense, your goal should be to stop them, not kill them. If you intend to kill them, then you're attempting murder. I don't care how heinous of a thing they are trying to do, if you inflict a serious injury with malice, it's attempted murder! In a self-defense situation, you shoot to stop a threat.
So, why would shot placement still not be important? Would you not want to shoot the bad guy in the spot that's most likely to make him stop? Well, yeah. But it's more important that you hit the bad guy, and if you think shooting at a big buck is unnerving, imagine shooting at a human when your life—not an empty freezer or big antlers—is on your mind.
The best way to obtain a hit in a dynamic, life-or-death situation is to aim at the center of the available target: center mass. This gives you the greatest potential to get a hit because you're aiming and shooting at the largest part of the target you can see.
The other thing that comes into play here is that, short of a hit to the central nervous system (the head or spinal column), handgun bullets are not powerful enough to be relied upon to instantly incapacitate an assailant. Humans collapse or die when their brain runs out of oxygen, and with a hole through both lungs this might take more than a minute. That's too long when you're fighting for your life.
The best you can hope for is that when you shoot the bad guy it will hurt him so much that he decides what he's doing is not worth the pain, or that he realizes he's shot and a trip to the hospital might be the next adventure he should undertake. If you can see your entire attacker, a center-mass shot also has the best opportunity to bring about this type of pain and mind-changing affect.
As important as shot placement is to hunters, high stress often clouds their judgment to the point that a center-mass shot is what they take. Hunters become the victim of buck fever and can't hold the gun still, or they think if they aim in the center it's the best chance they'll have to hit. This ultimately results in gut-shot deer that are often never recovered.
You can consider it a success if you shoot someone who is attacking you and they run off, never to be seen again. If that happens when you shoot a game animal, it's a failure. Hunters need to focus on lethal shot placement, and if the animal doesn't present a clean shot, no shot should be taken. If some troll is attacking you with a battle axe, you don't have the luxury of passing on the shot. You need to shoot immediately and you need to get a hit. You should also keep shooting and getting hits until the troll has stopped or turned to run away. Shooting at the center mass of the available target is the best and fastest way to get those hits.
But what happens when you've shot someone intent on killing you several times in center mass, yet they're still trying to murder you? Then it might be time to switch targets. The bad guy might be wearing soft body armor or, more likely, is on drugs and hasn't felt the bullets you've been putting through him. In this case, it might be time to transition to a head shot and try to destroy that central nervous system.
Truth is, nobody—including bad guys—likes to get shot. A couple hollow points shot at center mass, regardless of caliber, generally have a way of convincing bad guys that they're not messing with a "victim."