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how do you know what trades took place, and how many shares were traded, and at what price?Company generally has these records. Various other sources, e.g. PitchBook, compile them. In some jurisdictions (e.g. UK and India) they have to be publicly announced, though that's becoming less common.
TL; DR It is incredibly wrong to suggest private companies don't have a market cap. As in finance 101 wrong.
