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When I was kid, my siblings and I walked to school each day. There was an older man who always hung out on a corner along our route. Whenever we got near him, he tried to make small talk with us. He even started walking with us until we got across the street. Then he returned to his corner and waited for other kids to come along.
They called him a crossing guard. And I doubt that he was checked out any more than the PI on the OP's street. He was just the guy willing to stand out there and do that job for a couple bucks.
This seems like an example of how we have devolved to a state in which everyone is a bad guy unless he can prove unquestionably that he is not. The PI is doing a job, the cops have verified that (which is probably more verification that any of the neighbors have received), and he has done nothing to indicate that he is a threat to anyone. But he is still treated as a criminal because of what he might do. He might even be investigating one of your neighbors whose spouse suspects him/her of abusing their children.
If you feel threatened by him, then by all means take steps to minimize the risk you perceive by making the kids play in the back yard or playing along with them. Put a camera on the guy. But don't attack a man who has done nothing wrong.