They were moving through traffic with their blue lights flashing and sirens sounding on Station Street when the incident happened.
Witnesses say car parts were strewn across the road after the smash on Friday at around 3.30pm.
The drivers - travelling under the 30mph speed limit - then straightened up their vehicles and carried onto the emergency in Hucknall.
But the incident shocked many witnesses and several took to social media to say how lucky it was that no-one was injured.
Kirkby man Andy Warriner saw the crash and its aftermath.
The 29-year-old was sitting in the window of a cafe on Station Street when the crash happened.
He said: “One police car was in the left hand lane on Station Street and another police car turned right, so I think he realised he was in the wrong lane.
“He cut across the road, but another police car was coming down the road too with its lights flashing and it T-boned into the car.
“There was such a loud bang everyone looked to see what it was. Then they just drove straight off. Everyone was just gobsmacked it was the police.”
He said there were parts of the car left on the road after the smash.
“It was really busy at that point so it was lucky no-one was injured. We were amazed no other cars were hit,” he said.
A Nottinghamshire Police spokesman confirmed that two force vehicles were responding to an emergency incident in Hucknall when the crash happened.
She said: “They were negotiating traffic on Station Street in Kirkby when one police vehicle clipped the other. No other vehicles were involved or were damaged. As both police cars only sustained minor damage, and no officers were injured, they continued on to the job they were called out to.”