In Florida private individuals are legally able to buy/sell or trade firearms among themselves without problems as long as the two people are both Florida residents and neither is a prohibited person.
I did a trade a year or so ago just like you say. I found a guy on the local gun trade board that wanted to trade a pistol I had for a 12 ga. shotgun he had. We met at the parking lot of the local Super Walmart mid-day and did the deal. No problems. I will say that we filled out a simple Bill of Sale with the name, driver license number and description of each firearm with serial numbers for each person just for good business.
I'm guessing Texas probably has similar firearms laws.