This is a very interesting question, and here's how I do it.
First of all, we can have three people who all speak Chinese come and say a few random Chinese words. When you hear the different pronunciations from each of them, you'll basically be able to determine who's Chinese, and if you can determine who's who now, then you're fine.
If you still don't know who the remaining two are, then ask them to speak a few words in Japanese or Korean. This will naturally distinguish them. This is a very convenient method because our pronunciation is never going to deceive us about who we are. There's also between asking them to show their valid ID, which I think anyone in their right mind can think of, it couldn't be a simpler way, right?
There is also the fact that we can observe their behavior. In foreign countries, usually only Chinese people talk loudly on the phone. Japanese people answer the phone in a very low voice, very worried about affecting the people around them. Koreans generally bow when receiving help, all of which I have seen.