Short answer: No. For a purely private, in-state sale in Texas, there’s no legal requirement for the seller to record the buyer’s info if no FFL is used and no Form 4473 is involved.
What you still must follow (even without records):
Don’t transfer to people you know are prohibited (e.g., certain felons, minors, someone under a protective order, or anyone intoxicated), and don’t make interstate transfers without an FFL. See Texas Penal Code §46.06 and federal rules.
Good (optional) best practices:
Bill of sale / basic receipt (names, date, firearm description, price) kept for your own files.
Check Texas residency & age (e.g., TX DL/LTC) and trust your instincts.
Consider using a local FFL to facilitate the transfer if you want a background check and paper trail. The ATF publishes a short “Best Practices” brochure for private sellers.
Texas also has no general firearm registry (NFA items are different).
Not legal advice; just the general rules with sources.