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If you want to see the "magic" in real time, you can check the coordinator's publicly broadcast round ID hash: https://coinjoin.kruw.io/wabisabi/human-monitor
If a coordinator tried to cheat, you would be able to detect it since the hash in your client's logs would be different than the one you the coordinator announced publicly.
No, the coordinator doesn't.
How this works comes down to two things. First, a unique Tor identity is used for each input and each output, so they can't be linked at the network level. Second, some magic math: keyed-verification anonymous credentials. It lets the coordinator issue you so-called credentials; mathematical objects that you can then do interesting things with, like re-randomizing them or splitting them into several smaller ones.
The magic lies in this: when the coordinator receives your re-randomized credentials, it can verify that it issued them, but it can't tell which input they came from.
So throughout the whole process, the coordinator is essentially a blind man.
I want to know if I do a xoinjoin you know what's is mine and what's not in the output. Tell me how it works. Show me the magic. Don't redirect me to any link.