A seller who won't share the watch's serial leaves you unable to run an individual stolen-status check. That doesn't prove the watch is stolen, but you should not treat the missing number as a small detail or pay a deposit on the promise that it will arrive later.
Ask for a safe way to identify the piece
Explain that you need to match the serial on the physical watch with its paperwork and the result of a database search. A seller who worries about exposing the number in a public listing can provide a private photo or show it during an inspection. The key is that you can read the actual serial before you commit.
Do not accept the model reference in place of the serial. A reference can apply to many watches, so it cannot establish whether that particular example has a theft report. Nor should you search a number supplied for a different watch.
When the answer stays no
Pause the transaction. Keep the seller's messages and listing, and don't send money through a method that offers no recourse. If the seller later provides the serial, compare it with the engraving and documents before you search.
After you receive the number
A no-match result is one record of a search at a particular time, not a guarantee of ownership or authenticity. Check the seller, inspect the watch, agree on written return terms, and keep an invoice that identifies the watch by serial.
You don't need to accuse the seller. You do need enough information to check the watch you are being asked to buy.
Get the Serial Before Payment
Run a search only after you can match the serial to the actual watch and its documents.