Its possible to use a single transducer for transmit and receive - but this requires a transmit/receive changeover circuit to allow the transducer to both transmit high voltage signals and be connected to a sensitive receive amplifier. Vehicle reversing/parking sensors typically are like this.
Using a separate transducer for receive and transmit means the transmitter can be optimized for high power output and the receiver optimized for sensitivity, as well as isolating the receiver from transmitter and making the circuitry simpler (no transmit/receive signal switching).
Actually a "switchover" system isn't typically used; this is difficult to describe, but basically everything is connected together (the output and the input), and "buffered" via a couple of back-to-back diodes (reverse polarity to each-other in parallel - hard to describe, and looks even weirder in schematic form) to the receive amplifier. So - what happens when you ping is the receiver gets a "spike", but the diodes protect against this spike; since the system knows that it has sent the spike, it waits to "listen" for the echo after the spike. The main downside of this system is the fact that there is a "settle" time after the ping, so if an object gets very close to the sensor, the distance become immeasurable because it is in the "wait period", or is swamped by any lingering "settlement" noise.
On my other computer I had a link to this kind of circuit which was detailed on a ultrasonic transducer manufacturer's site; if I can dig it up later I will post it here (that, or you can look it up on one of my other older posts - I know I have posted about this before).