For me on PC using the Xboix 360 controller its left joy stick pushing in. You have to be moving though as if you are parked that will shut the car off as per the usual. Start driving the car then push in on the left joy stick and that should work for you. To sound the siren use RB trigger on top of the controller.
For other controllers I will guess the same way you shut the car off is the same way you turn on the police lights WHEN DRIVING. Again if you are not moving then it will shut the car off.
Hope this helps. Peace. !