I watched the driver on either side of me this morning on my way to work. We were stopped waiting for a red light and I could see that both of them were totally focused on the that light. Neither one of them moved their heads and immediately started off when the light turned green. What a wonderful example of blind faith!
Yes, cross traffic is supposed to come to a stop when our light changes to green. You can probably get by believing that they will until someone makes a mistake or wilfully disobeys the light. Blindly hitting the throttle as soon as the green is exhibited doesn't seem like such a good idea now, does it?
Both of these drivers should have noted that their light turn green, looked left, looked right and insured that cross traffic was coming to a stop. Having done that, they should look left again and then moved their foot from the brake to the accelerator if the way was clear. It never hurts to supplement faith with a bit of caution.
This same practice of looking left and right should also take place before you enter an intersection where you already have a green light. Similarly, if there are no traffic lights and cross traffic has a stop sign, look left, right, left again.
Finally, you don't have to be a driver to exercise the left, right, left scan. A pedestrian stepping into a crosswalk would be wise to do this before leaving the curb. Many drivers don't like to stop before turning right on a red light and are probably looking left for cross traffic rather than looking right for you.
Observing at Intersections - Page 62, Road Sense for Drivers