It had nothing to do with socialization or training. The Pekingese was simply more aggressive and was never listening to our command, when outdoors at least. And at some point we simply have up on trying to educate him, because that was a losing battle. The Rottweiler on the other hand knew what we wanted at all times. She even knew when we wanted her to become aggressive without any special words spoken. Obviously she was never aggressive towards the Pekingese or she would have killed him. A hard thing to do, since this kindness wasn't returned by the Pekingese.
Now don't get me wrong, I loved both. But the Rottweiler was clearly much smarter and more adapted to live as part of a family ;-)
PS: Broke my heart when she died, hence why I'll never adopt a dog again. They live much shorter lives, hence you get to be a parent of sorts, then watch them die, sometimes horribly.
It is not unusual to see ill-tempered small dogs at all.
Without a proper control, I put it down to environment and how they're treated. Many small dogs are "purse dogs" that are treated like accessories instead of dogs, and I hypothesize that has something to do with it. I mean, people get a small dog because it fits a certain lifestyle, and that lifestyle may not include treating your pet such that he gains confidence in all situations. But that's just me talking out my butt, I have nothing that isn't anecdote or SWAG to prove my point.