That post is close to correct, a handgun purchase requires the purchaser to have a handgun (firearms) purchase certificate or meet one of the exceptions. The issue is the statute shown is old a CCW or police certification now also applies as an exception.
Current Law:
http://uniweb.legislature.ne.gov/laws/statutes.php?statute=s692400300069-2403. Sale, lease, rental, and transfer; certificate required; exceptions.
(1) Except as provided in this section and section 69-2409, a person shall not purchase, lease, rent, or receive transfer of a handgun until he or she has obtained a certificate in accordance with section 69-2404. Except as provided in this section and section 69-2409, a person shall not sell, lease, rent, or transfer a handgun to a person who has not obtained a certificate.
(2) The certificate shall not be required if:
(a) The person acquiring the handgun is a licensed firearms dealer under federal law;
(b) The handgun is an antique handgun;
(c) The person acquiring the handgun is authorized to do so on behalf of a law enforcement agency;
(d) The transfer is a temporary transfer of a handgun and the transferee remains (i) in the line of sight of the transferor or (ii) within the premises of an established shooting facility;
(e) The transfer is between a person and his or her spouse, sibling, parent, child, aunt, uncle, niece, nephew, or grandparent;
(f) The person acquiring the handgun is a holder of a valid permit under the Concealed Handgun Permit Act; or
(g) The person acquiring the handgun is a peace officer as defined in section 69-2429.
For long guns the certificate is not required by law but allot of sellers require it just so they can be certain they are not selling to a person who is barred from purchasing.