You should check the data of the transducers that are available for you, and see, what's possible with these. There are considerable differences in terms of operation frequency, bandwidth, directivity, impedance. Separate transmitter/receiver designs allow an easier signal processing and are better suited for continuous (CW) operating doppler sensors. Single transducer designs are mostly used for radar like time-of-flight sensors. But they are less sensitive for motion detection applications, I think.
Ultrasonic motion detectors have been used as burglar alarms before PIR sensors came up. But they are rather succeptible to false alarms, e.g. caused by movement of the air.