Any mode S transponder is transmitting any time it's in GND, ON, or ALT, regardless of whether the engine is running or not. If it's got power and is selected in a mode that transmits, it's transmitting.
Just turn it on and watch the reply light. It'll blink, even on the ground.
Generally, unless you're at an airport with ground surveilance radar, ATC filters unwanted returns off the scope (I forget offhand if it's by altitude or by airspeed or some combination of both).
I also don't know of any modern unit that doesn't do some sort of basic internal self-test when the power is applied.