It completely depends on the city.

Some buttons trigger the “walk” sign but don’t actually change the traffic signals for cars.

Some buttons are placebo, perhaps with a secondary function of emitting a loud tone to let blind people know the walk signal has turned on.

Some buttons trigger the traffic lights to stop.

Some buttons turn the traffic light to a 4 way red light.

I have encountered all these types of buttons on the east coast of the US in various states.

Around here we have a wide variety. Sometimes they do nothing. Sometimes they don't change the main traffic pattern at all, but won't ever display "walk" unless you press them. Sometimes they change the traffic pattern. Sometimes behavior changes at different times of day. Sometimes they're legit broken - at one intersection, a button I used only worked maybe 30% of the time, unless you were holding it down at the exact moment the cycle changed, then it worked 100% of the time.

I recently stood at a light for like 15 minutes before I realized a button was available. There was no pedestrian crosswalk scheduled at that light at all. Once pressed traffic in all directions stopped in about 1-2 minutes. It was glorious.