IME/IMO, it's mostly genetics.
I've seen any number of pitties come out of the shelters around here that are socialized like crazy with other dogs from day one (sometimes appropriately, sometimes less so, but it doesn't seem to matter that much, tbh). They hit social maturity around age two, two and a half, it's like a switch gets flipped and bam!, suddenly that is most definitely not a dog park dog anymore.
(For the sake of clarity, I want to note that I'm not saying all pitties are like that, not by a long shot. Many of them stay fine with other dogs for their whole lives. But our city shelters do get a lot of others that aren't, and that's not always evident in baby puppies of unknown lineage.)
As with a lot of other things, the genetic influence is what Patricia McConnell has called a "blueprint written in pencil" -- how the dog is raised, early socialization, pleasant or unpleasant experiences with other dogs, etc., can and will have a very strong influence.
But if the dog comes from parents and a long line of other ancestors who were not DA, then that definitely stacks the deck in your favor.
ARCHMX TDCH Pongu the Insane, CD-C, RE, RL1X6, RL2X5, RL3X2 (GSD mix, b. Apr 2010)
Crookytail the Tigerwuff, RL1, ITD (Akita mix, b. Jan 2011)