We are medium sized privately held bank and we do not deal with many Non-US persons. The ones we do deal with are primarily students, but they tend to go with the larger banks, and the ones we do deal with we require a copy of their student ID, Student Visa and Passport. For the non-students Non-US persons we require a copy of their passport, work-visa, green-card etc. We've done this even before CIP.
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The opinions expressed are mine and they are not to be taken as legal advice.