They did. Some of it even pretty sophisticated (They had the first ever scanned array Radar, where you didn’t need to rotate the antenna to rotate the beam) The history of Radar is a history of lots of inventions being made in parallel pretty much at the same time by different people who didn’t know about each other. For quite some time, technologically, it was an arms race where both sides were pretty much head on, until the Allies managed to pull ahead by inventing the cavity magnetron (allowing drastic downsizing of very powerful Radar transmitters), and bringing US mass production capabilities into the game.

The Battle of Britain famously being won by Radar, for example, is only a very superficial part of the story. What actually won it wasn’t Radar alone, but much more the sophisticated system of command and control behind it, turning the data obtained from Radar (and other methods of aircraft detection, such as old fashioned visual observation) into workable interception orders for fighters pretty much in real time. In fact, the Germans didn’t think of British Radar as much of a threat at the time, for mainly two reasons:

1. They had Radar themselves and didn’t find it all too useful for air defense, because they lacked a sophisticated system of command and control to process the data obtained by Radar in real time.

2. They had captured some mobile British Radar sets during their conquest of France, examined them, and found them to be rather crude and less capable than their own.

The chaff from the original post also has a particularly interesting background story: Both sides had invented it independently pretty much at the same time, and then both sat on that invention for an entire year, not daring to use it operationally for fear of the other side copying it, because it’s really quite simple to make once you know about it.