Now that I'm a driver in LA, I totally wait for others to start moving, in case there's a crazy driver who wants to run the light at 40+mph. I even consider the possible cushioning that other cars might give me if I'm in the leftmost lane and a crazy driver ran the light coming from my right.
Going left when there's no green arrow is also dicey at times. You have to trust that the people coming at you will slow down at a yellow light so you can go instead of accelerating so they can make it. When it's really busy I just usually relegate myself to turning as the light turns red because I just don't know if the people coming at me will stop.
I've also almost been hit on a left arrow because a driver in the other direction thought it would be a great idea to turn right into the middle lane...which is where I wanted to go. Driving in SoCal will teach you how to be a defensive driver really really fast!
You definitely see people running left arrow lights regularly here, especially on freeway onramps.