Not trying to bump up an old thread, but as someone who isn't from CT but whose better half does have family there and do visit every once in a while (its a long drive from NC)......
What kind of permit are we talking about? an actual carry permit or like a range safety card?
Most public ranges around here (NC) make you take a short saftey quiz (20 minutes or so) to prove you know the difference between a muzzle, chamber, grip etc.
Have never heard of anyone needing anything from the state to shoot at a public range