Our minds bias towards telling ourselves that we're okay, most of the time, and we rationalize constantly in order to get through the day, imposing our storytelling techniques on capricious events.
That means that we behave "good" just to the extent that we're aware that we might feel guilty and not much farther - and in that sense, we can train ourselves into greater moral fortitude with deliberate practice in empathy, exposure to diverse views, and development of a personal ethical philosophy. I don't see "personality" as crucial to that - it can differ - and changes in environment and status have a huge impact on what we perceive as acceptable, as well as medically diagnosed conditions that literally disable our ethical senses. Across a population, what is probably most important is to have a little bit of wisdom distributed among everyone so that we aren't burdening each other constantly. "Crossing the line" into violence happens after a huge, huge number of other failures have already taken place.
Systemically, we aren't even to the point where we might discuss this as a problem, though. We don't yet see wisdom as something everyone needs in the same way that we think that everyone should be fed, housed, and clothed: If you have those things and you're unhappy, the systemic recourse is to engage in consumer activities marketed around your perceptions of unhappiness.