If you are within earshot of your children, you are not "letting them play by themselves." You are supervising their play from a distance. They are not the same. Know why bullies have it so easy nowadays? Because once upon a time, a bully was usually a kid who came from an abusive household and they were just demonstrating what they learned at home. Eventually, they'd end up getting their asses kicked by a kid who had the means and background to defend himself, and they'd learn something from it.
Today, the world is full of little Eric Cartman wannabes: little asshole kids whose parents hover around them and will defend them to the ends of the Earth. It's screwed up child socialization and it's ruined childhood education because teachers with three degrees would rather do anything than take abuse from those bad kids and their pathetic, enabling parents.