Whether or not the vehicle giving the jump should be started when giving a "jump" start depends on the vehicles involved.
As Bill says: lots of late model cars need the extra couple of volts from the donor car's alternator spinning to activate their sensitive electronic ignition.
OTOT, a vehicle like a diesel with glow plugs and no spark type ignition might start just fine with a 12V jump.
Also, some vehicles make poor "donor" for jump starting. I've known of some to have their electronics get fried by the load of starting the other vehicle, or the sudden current inrush of the vehicle being started suddenly surging a large amperage charge when it's high powered alternator comes on line.
Sure, I've jumped all sorts of 12V vehicles from one another... Using a motorcycle to jump the lawn tractor; a car to start the big tractor etc... I've even successfully used a motorcycle to jump start a car; by allowing the cycles generator/engine some time hooked up and running to get the car's battery partially charged.
But everyone should be aware that not all vehicle electrical, and certainly not their charging systems, are compatible. There is always the possibility of damage to one vehicle or another when "jump-starting" vehicles with cables. I'll give this hint though: sometimes it helps to run the "ground" cable from frame to frame instead of from battery negative to negative. But that's still no guarantee. Hate to say it, but when I'm in a newer vehicle and some poor citizen is wandering around a parking lot looking for a jump start; I usually beg off. Back years ago, even in old 70s cars and trucks with an alternator, I'd gladly offer to help someone get started. But nowadays, I'm not risking thousands of dollars in wiring and computer crap in my vehicle because someone else doesn't have an auto club card to get a free jump.