Alan Snyder admitted in the Season 1, Episode 6 entitled "Yoknapatawpha" that he's not able to tell the drones what to do. He didn't say who was able to do so. It may be that someone in Homeland Security or higher in the Transitional Authority is able to call them, or possibly only The Hosts. The third possibility is that the drones control themselves and are The Hosts, or that they work with them.
While they are used mostly for surveillance they are armed with an automatic weapon and are also used to kill people and to shoot down human aircraft.
In the "The Garden of Beasts", it was revealed that the Blackjacks are able to control the drones. A human was seen using a laser to mark, or ("paint" / "designate"), a vehicle as a target. The vehicle was immediately destroyed by a drone attack.
The Drones appeared after the Arrival.