um, no, they're not public roads, but they are classified as one and the same.
a public road is classified as anywhere that is easily accessible by a car (i.e. you can drive straight off the street into the carpark, its a sealed road, blah blah blah)
and no, i think you'll find that if you teach a kid (without a license/learners) in a shopping centre carpark, you'll get reamed just as hard as if driving unlicensed on the street
and yes, police DO have jurisdiction over everything, and everywhere..
hell, they can fine you for doing stuff in your own backyard!