A Tudor Black Bay's model name won't tell you whether that specific watch has been reported stolen. You need the serial for the individual piece. Ask the seller for a legible number before paying.
Compare the serial at each step
Match the engraving with the card, receipt, and service records. If the number is hard to reach or read, ask a watchmaker. A search on a listing number is useful only if the delivered Black Bay carries that same number.
Interpret the result
A database match should stop the sale while you contact the provider. No match means the records searched had no corresponding report at that time. Save the date and result with your purchase file.
Check authenticity separately
Theft status and authenticity are different. A copied serial can look plausible, and a genuine Black Bay can still be stolen. Use an independent inspection if you need an opinion about originality.
Check the Black Bay's Serial
Search the individual number and keep a copy of the dated result.