The spy device was no "resonant mixer"! It was a microwave resonator with one wall used as an acoustical membrane. If someone talked in the room, the voice modulated resonant frequency.
The spies across the street installed a microwave transceiver sending a CW signal; the returned signal was mixed with it, and the IF output from the mixer was amplified into a headphone. The operator could listen this way to conversations in the room across the street.
The principle is similar to a "solar phone" used a century ago; a mirror was set to reflect solar light across a street where it hit a simple resistive light detector connected to a headphone. Talking to the mirror modulated the light amplitude.