I'm not certain about FATCA, but I can confidently say that PayPal and U.S. banks report in great detail to the tax authorities. As mentioned in another thread, a close friend in Denmark is currently facing severe challenges with the tax authorities.
I am closely following the entire process to learn from it and to contribute information here on OCT that could be crucial for others in the same situation, as well as for those who are simply interested in the subject.