I can describe how I test ultrasonic transducers:
As a receiver I connect a 40 kHz or 24 kHz device to an oscilloscope.
I use a 555 as a tunable ultrasonic generator and connect a tested device to its output. As I tune the generator around the expected frequency, I can see on the scope the received voltage when the both devices are located close to each other. By tuning the voltage to a maximum I can check the resonance frequency. By rotating one device against the other I can see how directional it is.
For high-power devices I used a 3W transistor amplifier with an output transformer tuned to 24 or 40 kHz as needed. The voltage on the transducer was ~100V p-p and the receiver can be separated several meters to see the signal.
With high-power ultrasonic devices be aware that they should be immersed in water (for cleaning), or attached to a resonant plate. Also a good heat-sinking may be important.
For other frequency than 24 or 40 kHz I can use a small electret microphone from a cell phone or a tape recorder. It is not resonant and only operates with 3 to 12 V positive power supply connected through a 10 kOhm resistor, to supply the internal FET preamplifier. Old-style crystal microphone or phono pickup can also be used.
Do not expose yourself or other persons to high ultrasonic power, the hearing can be damaged. Use tested devices as above to check transducer operation. Pets like dogs and cats may be disturbed by ultrasound, too.