I didn't read the theory of operation at the leang website that you list, but IMO, you should be seeing something like this:
-- A 40kHz drive signal across the TX transducer for the duration of the 200us PING signal. A 40kHz signal has a 25us period, so you should get about 8 cycles of drive at 40kHz into the TX transducer for each PING. The 555 frequency should be tuned to the resonant frequency of the TX transducer. You can hand-tune the frequency in the next step...
-- Put an RX transducer directly in front of the TX transducer, and only connect your oscilloscope across the RX transducer (nothing else yet). Maybe separate them by 1mm or so. As the PING signal pulses, do you see a small signal on the RX transducer? I don't know how big the signal will be with only the oscilloscope input amp to magnify it, but I would think that you should be able to see it with such a small TX-RX separation. Worst case, you can touch the TX and RX transducer faces, but that will make the fine tuning of the TX frequency less valid. Assuming you see a small RX signal at 1mm spacing, fine tune the 555 frequency to maximize the amplitude of the RX signal. That fine tuning will help to maximize your sonar range.
-- Next, reconnect the RX transducer in your Receiver circuit, and watch the output of the first opamp stage. It should be an amplified version of the 8-cycle sine wave signal that you were seeing on your oscilloscope when you watched the output of the RX transducer alone. If it is not, then there is something wrong in the first stage amplifier circuit. If it is working right, then your gain will be set to barely get a good tone through the first stage when the reflecting object is at max range. With the gain set like that, the output of the first stage amp will be saturating into the rails when the reflecting object is very close. You might want to look into putting a soft saturation clamp circuit on the first stage to be sure that it doesn't slow down too much to keep up with return echoes from close objects. I'll leave that part as an exercise for the reader. Also, verify that the return echo of about 8 cycles at 40kHz (it may take a couple cycles for the RX transducer to ring up to full amplitude) get smaller in amplitude and more time-delayed as you increase the spacing between the TX and RX transducers (or increase the distance to the reflecting object). Remember that the speed of sound is about 340m/s at sea level, so expect about a 3ms delay for each meter of separation (ignoring circuit delays and ringup time factors, which you will need to calibrate out of your final answer).
-- Assuming the first stage is working okay, trace the signal through the 567 tone decoder. The 567 should provide some noise immunity against other noises that the RX transducer picks up. You should tune the 567 to be most sensitive to the frequency that you found from the earlier tuning that you did to maximize TX-->RX throughput. I've never used a 567, but it looks from the circuit like it will output a pulse when it sees a tone that is the same one it is looking for. I don't know how many of the 8 cycles it will take before the 567 drives its output, but you can probably figure that out from its datasheet.
Have fun!
