I think the answer is consistency. If you set firm, consistent, and meaningful cause/consequence relationships and enforce them predictably, your child will eventually learn. Trust yourself.
As to the comments, I'd take them to heart only if they're offered as constructive feedback from a parent whose child has an altercation with your child.
As long as you're sincerely addressing the root cause of the behaviour and doing your best implementing a reasonable approach to correct it, I would let those comments roll off my back. By the time they're parents of 3 year olds, most other parents should have the sense to know that parenting is more art than science. There are inevitable lessons that have to be learned along the way, and that involves some friction.
FWIW, I had to leave a store yesterday while my son (17mo) threw his first public tantrum. He didn't want to stop playing with a toy that I was about to buy him so, after a failed discussion, we didn't buy the toy and left. The fuss almost certainly made others uncomfortable. I received a few glares from people who were clearly sympathetic to my son, but I am confident that my quick reaction sent a strong message to my son about acceptable behaviour. I'm the parent and I determine our standards of comportment, not a bystander.
