Their behaviour is programmed based on the characteristics of the intersection. If it's a very lopsided one where one direction has heavy traffic and the other very little, then the button might trigger a light change.

Other times, the light timing doesn't change, but the button will trigger the crosswalk to indicate walk after the next change.

Other behaviours I've seen is it changing the advance green/left turn signal, or showing the walk symbol while all traffic is stopped.

And then I've seen a few that do appear to do absolutely nothing - but that doesn't mean that's the case. For example, an intersection might have a button that only does something at night when there's very little traffic, but doesn't during the day when the intersection is constantly busy.