People don't really need to share the "belief" or have any intellectual understanding about it. Eventually there just becomes no real incentive to be violent.
Right now (well, even besides joining the government in the first place lol) someone who wants to be violent benefits a lot from the government prison/punishment system. So they go sit in a cage for a few weeks or a few months or something. In a free society governed by peer to peer reputation, a rapist or a thief will be committing social suicide and wouldn't just return smack dab to the middle of polite society like that.
So the average person doesn't need the intellectual understanding of non-violence, they just will have less and less reason to ever be violent, and it snowballs.