They CERTAINLY do not in California. I was told they would do so, but in fact, the DOJ has the records. My brother in law is CA DOJ, and I have had a gun stolen, so I know that the DOJ keep these records.
The answer to that is supposed to be a stolen guns database. It only contains records of guns known to be stolen. It’s a good idea to keep the serial numbers of your weapons for that reason. In theory you should be able to recover a stolen gun if the cops recover it.