From the BATF NFA Handbook, Section 3.7:
Section 3.7 Maintaining registration documents. A person possessing an NFA firearm registered as required by law must retain proof of registration, that is, the document showing the person’s registration, which must be made available to ATF upon request.
It does not specify whether the document must be original or copy. However, it is simple for BATF to determine rfom a copy that a firearm is indeed registered to the person named in the document.
Also, from Section 3.4.2:
3.4.2 Prohibition on ATF’s disclosure of tax returns or tax return information. NFA forms are treated as tax returns and registration information in the NFRTR is considered to be tax returninformation. ATF is generally prohibited from disclosing tax returns and tax return information. However, firearms registration information may be disclosed to registered owners/possessors of the firearms.