If you are speaking of the Pistol Training class feature gained from the Pistolero gunslinger archetype, it functions different from the Weapon Training class feature of the Fighter class, so yes, I'd say they stack.
I was under the impression that, as the Gunslinger is an alternate class of the Fighter, you can't have levels in both. I'm not familiar with any other way to get Weapon Training in a firearm.
Ah, didn't notice the change between the gunslinger playtest and the Ultimate Combat printing that the gs is no longer an alternate fighter. Interesting.
Yes and it's a fairly common build. 5 levels of Pistolero and then go straight fighter for the rest. Gun training gives you dex to damage, weapon training gives you bonuses to attack and damage as you level. Pretty simple.