The way I understand it, it works in the following manner:
Long guns at a gun show from a private individual: Background check and state paperwork required.
Long guns at some other private party situation (ie meet a buddy from THR who is also a CT resident to buy a rifle, and he is not a dealer): As long as the seller does not know or believe thebuyer to be ineligible, everything is fine and dandy.
Pistols and revolvers always require paperwork and a call-in check.
I advise verifying the info with the Special Licensing Division.