I was wondering if Sthil has a stolen saw announcement that they send out to their dealers. If not they should. It would be easy to contact Stihl and have them brodcast description or serial numbers of stolen equipment to all their dealers. As insentive they might offer a bounty on any recovered equipment. I just lost my O36 serial number 231740332. I bought it new so I know it registared to me and one dealer I contacted and confirmed I was the listed owner. I replaced it with a 361 but would love to recover the O36. I'd be happy to pay a bounty on the return of the saw and a bigger one on the prick that took it.
While that isn't a bad idea, it really wouldn't do much as for getting your saw back. First off, barely any dealers take trade ins. Second, most saws that are stolen are running saws, no need for service. Third, you would have to be really stupid to try and deal a stolen saw to a dealer which probably just sold a new saw to the guy who got robbed.
Dealers do look at saws if something is fishy. All stihl saws have the serial number marked inside of the case. So if the serial number is scratched off, then the deal should split the case. My dealer has done it a few times.
A bulletin board at the dealer with descriptions of stolen saws would be a good tool to get the word out.
Hopefully you find your saw. I like the 036 better than the 361.