Both sides had radar in general, and knew about the other side having it.

The carrot propaganda part is true though, but it is specifically about covering up the fact that the allies had developed aircraft intercept radar, i.e. radar mounted on the planes.

https://www.smithsonianmag.com/history/carrots-cant-help-you-see-in-the-dark-heres-how-world-war-ii-propaganda-campaign-popularized-the-myth-28812484/