The YAK 52 aircraft crashed in a field near Cooksmill Green in Essex.
Firefighters called to the scene reported that "the plane was completely alight" when they arrived.
The plane had taken off from the nearby North Weald airfield. Both victims' next of kin have been informed, police said.
Instructor Simon Hulme, 33, and his 43-year-old student Spencer Bennett, were killed in the accident. They were on the last day of a three-day formation flying school course when the aircraft plummeted from 1,800 ft into a lake, British media reports said.
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"It's completely covered by tarpaulin and there are about eight police officers in attendance."
Sam Gildersleeve, from Ongar, was having lunch with friends when he saw the plane flying low.
"At this point we all looked at the sky and the plane attempted to do a somersault unsuccessfully, we heard a popping sound and noticed the plane did not re-emerge," he said.
The Air Accident Investigation Branch has sent a team to investigate the crash.