I ask sellers for serial numbers daily in registering Kay basses and cellos and building the Registry data set, which expands our knowledge of the instruments and offers services to owners, restorers and retailers. Reliably 10-15% of sellers don't respond to these requests for whatever reason, the lion's share of them in the Boston-DC corridor, so it looks like one factor may be a cultural bias.
For what it's worth, on the rare occasions where a registered instrument was stolen and rediscovered with our help, the legitimate owners got almost no help from law enforcement in recovering the instrument, and huge hassle trying to get the person holding it to give it up. (It's never the thief of course, it's someone who bought it from the thief or a third party and is trying to sell it legitimately.)
Given this experience it's my feeling that it's highly unlikely that an illegitimate owner would disclose a serial number, due to the risk of being caught at it, highly likely that anyone disclosing a serial number has nothing to hide, and that the likeliest result of using a serial number to try to gain the instrument illegitimately would be quick charges of fraud and whatever local legislation frowns on using cops this way.
I really don't get why people worry about serial numbers, it's nonsense.