I came up with a theory; child imagination brings them to life. Going by the unified Pixar theory, we know that children's emotions have some sort of magical power as laughter and screams can power monster equipment (or rather, mutants in the far future eventually invent technology that is powered by the laughter and screams of children).
Just about every toy takes on the personality of whatever they are played as. To start with OP's question, sentience would have to happen at some point in the toy store. Every toy we've seen from a toy store behaves like the intellectual property they are; Barbies act like girls and Buzz Lightyears act like Buzz Lightyears. And in an animated short, we see some kid buy some dinosaur toys that didn't realize they were toys until the end. They get this from all the kids that go in the toy store and imagine the toys. Since they're passing through, they imagine the toys at face value, and as such the toys adopt that personality. This also explains why the Buzz we know and love believed he was real for so long - when Andy plays with him, he probably pretended he was Buzz Lightyear, so Buzz kept getting this personality reinforced. That was until Andy lost both Buzz and Woody, and he realized how much he missed them both and played them both at the same time. Army men get played with pretending they are army men. Andy never saw Woody's Roundup, so never pretended Woody as that and just pretended him as a generic cowboy, which is what he himself came to believe himself to be. Stinky Pete was never played with by a child; his only owner was a toy collector who saw him as a commodity. Which is exactly what he became and aspired to be - a commodity (as seen in Monsters University, adults have magic too, it's just monsters that lose this power for some reason).
It also explains why some animal toys are sapient and can talk while others can't. Andy imagines Hamm as a talking villain, while he imagines Bullseye as a mount for Woody (and it's likely Bullseye's previous owner did the same). Hence Hamm can talk and Bullseye can't. Sid's baby head was likely pretended to be some sort of sci-fi monster while Big Baby was likely played with and pretended to be a baby. Hence why Big Baby behaves like a baby, and that thing in Sid's room doesn't.
Now, that leaves us with a seemingly glaring flaw. Hamm and Potato Head aren't villains, but Andy imagines them so. While they aren't villains, they're definitely the biggest assholes, and they both trust Woody the least. As for why Lotso turned evil despite the fact that Daisy almost certainly never pretended him to be as such; remember he was her favorite toy. That put him a leadership role similar to Woody. And he didn't turn full evil until he went to the daycare. A pink teddy bear that smells like strawberries? Yep, that's gonna be a favorite, so he quickly rose in rank and the power went to his head.
TLDR; it happens in the toy store.