My personal feelings, as both a buyer and a seller FTF is that I will prove to you that I am a same state resident as you by SHOWING my driver's license or military ID + orders to you. I will tell you I am not prohibited from purchasing a firearm. I will sign a receipt for your money or for your gun [strike]and that receipt may certainly have my address and phone number printed on it[/strike] - that info is in the phone book anyway.
You will not get, nor will I require, to see a copy of a carry permit. A carry permit is a tax paid to a state for the privilege to conceal a handgun. I don't expect it to be used in a private transaction nor for any purpose other than concealing a handgun, because I won't support the state infringing upon persons' rights to possess firearms any more than is necessary by law in order to exercise that right.
You won't get, nor will I require, a copy of my driver's license, military ID card, DL number or Social Security number. We are just private citizens. I likely don't know you from Adam and you don't know me. That's just foolish to give information like that to a stranger.
In Texas, no paperwork is required at all. A signed receipt (bill of sale), just like for any major item being sold, is a good idea, in my personal opinion. Both parties are free to walk away from the deal at any time up to the point of after the exchange of money/firearm if they don't like what they see/hear.