Oh, yes, they push the buttons. Then they push them again, and again, and again. I've seen dozens of people do that and get more and more irritated when the "Walk" light didn't appear at their command.
Traffic lights are programmed to change at certain intervals, and pushing the button won't make vehicles stop for the convenience of a pedestrian. Instead, it will make the "Walk" light appear at the appropriate and safe time in the sequence, usually after the left-turn light finishes sending cars through the crosswalk and vehicles going the same direction as the pedestrian get the green light.
No matter when someone pushes the button, the signals will change when they're supposed to do so.
It's only necessary to push the button once.