At least where I live, the non-placebo buttons have an immediate effect - press, and the walk light will change to a blinking don't-walk hand, which then turns to solid as the corresponding traffic light turns yellow/red.
That's not how it works generally (that only works at low traffic intersections). More typically, the walk button registers that there is a pedestrian wishing to cross, without which the system skips over the pedestrian phase of the traffic cycle. This is less important in big cities where it is assumed there is nearly always a pedestrian waiting to cross, which is presumably why they were disabled in NYC. However, in this mode of operation the only way to test whether the button is functional or not is to not press it and see if a full traffic cycle is completed sans pedestrian crossing.
Incidentally, I worked in a building this summer with a working door-close button in the elevator, and it was immensely satisfying.
