I STRONGLY recommend exchanging information between buyer and seller for the following reason: The gun may be stolen, or otherwise linked to a crime.
Here is a common scenario for you: You are pulled over for any lawful reason and the officer, for whatever reason, decides to check the serial number of the weapon hou have in NCIC. Of the weapon is reported stolen, you will probably have some 'splainin' to do. Personally, I would prefer to be able to tell the DA or the investigating officer that I purchased the weapon on (insert date) from (insert buyer) for whatever sum. If the want is for any date prior to yhe purchase date, you will have a much easier (and cheaper) time getting off the hook. The reverse can also apply if you sold the gun to someone and it is later used in a crime. If you purchased it from a dealer, there will be a 4473 on file. The National Tracing Center (ATF) in West Virginia will try and track down the last known papered owner of the gun. If you are that person, it would be REALLY HELPFUL in your situation to have the buyer's information available when they come a knockin'.
I investigate identity theft as part of my job, and I do understand this concern. If you are getting bad vibes about a transaction, RUN - DON'T WALK away from it.
With anything that can come back and bite you in the ass - C. Y. A.!!!