You are allowed to give gifts of firearms as long as they are to me.
More seriously, there are no real NYS laws dealing with private transfers of rifles and shotguns. Federal law tells you that you can't transfer a firearm to a resident of another state, or a known felon, etc. But NYS does not impose any additional requirements on private party gifts/sales. (The foregoing does not apply to New York City residents, who have their own rules.)
As far as pistols, permit paperwork is required to take the pistol off your permit and put on another's permit. How that is done depends on the county. Here in NYC, to buy a pistol in a private transfer we are required to obtain a notarized bill of sale from the seller as well as a special letter from the NYPD.
Different rules on pistols apply to POs, who don't have permits.