It's based off a single chip, the TDA2822M, which is a small power amp "intended for use as dual audio power amplifier in portable cassette players and radios": TDA2822M.pdf
I'm not really an expert on this, but here are some notes:
1) Although it has an inductor before the power supply, presumably to block AC noise coming from the car, it's case is plastic with no shielding. Also, although it has pretty good quality cables going to the headsets, the braided coax shields are not used (they are cut off). I think that is what makes it so susceptible to alternater/spark whine. I suspect that lining the case with aluminum tape and grounding that, and hooking up the braided shielding on the cable would help a lot.
2) The power supply chip, an L7812CV tries to put out 12v, but doesn't drop out until 2v. This means you can run it off a 9v battery just fine, it just puts out a lower voltage in that case. I'm not just getting this from reading the datasheet - I've actually tested this.
3) The TDA2822M will also run just fine down to about 2v; it just puts out less power to the speakers at lower voltages. It's not linear either - at 12v it'll put out about 1.5W per channel (through the 16ohms of the Terraphone headset speakers) but at 9v it's down to about 800mW.
4) I'm not positive that diode D2 is a 5.1v Zener, but it's the only thing that makes sense. I believe that's the phantom power for the mic. If my calculations are right, it should be providing about 5mA through the zener and giving each mic about 1mA.
5) I think resistors R11 and R12 not only provide a path for the phantom power to get to each mic, but they're also where the mixing is done. In other words, it looks to me like your own mic is attenuated by what is effectively a 10K resistor before getting fed to what eventually becomes your output channel. The other person hears your mic at full bore.
6) I can't see any reason for caps C7 and C8 other than to block the phantom power DC.
7) I have no idea what caps C9 and C10 are for.
8) The radio input is set up so that the radio sees the 47 ohm resistor; I presume that's for impedance matching. A small amount of the rest of its signal is peeled off and sent into each channel. There is likely some feedback from the mics into the radio, but since the mic signals are small it's probably negligible.
9) The rest of it is almost directly out of the TDA2822M datasheet.
10) The TDA2822M is rated from -40 to 150C. The L7812CV has a storage range of -65 to 150C, but an operating range of 0 to 150C, which could come into play at colder events.
I don't see anything particularly wrong with the circuit. It's very simplistic, and the boards are pretty clearly homemade, but the traces are good and the soldering seems fine. I mean, it's about $10 in electronics and maybe $40 in the cables and case and so on, and if I were them I would've put in some preamps with some spectrum filtering and used a better power-amp, all for maybe $5 more, but hey. I'm just reporting what I found.