Shooting them down from the ground is next to impossible. They don't hover around waiting for someone to come by with a shotgun in their hand, catching them by land (ie. chasing them in a car) is not feasible.
Just to give an idea how hard it is to hit airborne targets from the ground with traditional guns: I once spent an afternoon shooting at a slow moving fixed wing target drone with tracer rounds from a 12.7mm anti-aircraft machine gun. There were about 50 of us taking turns, each with a few hundred rounds to shoot at the damn thing and the target aircraft didn't get a single hit.
My guess is that the drones are conducting signals intelligence, listening to radar signals and radio comms around sensitive installations (airports, military bases) and surveying the response time to a sighting.
