Do stolen share certificates belong to the thief?

The shares themselves still belong to you, but you no longer have proof of ownership, which you would need to sell them on. It's no different from the title documents (US) / V5 (UK) for a car.

There is lots of relevant information here including what to do if you have had share certificates lost or stolen:

Each company’s procedures may vary. However, there are some steps that the shareholder must follow. First, the shareholder must describe the loss and any facts surrounding the loss in an affidavit. Second, the shareholder may be required to purchase an indemnity bond. The purpose of the bond is to protect the corporation and the agent in case the lost certificate is somehow redeemed by another party at a later date. (Think of it simply as additional insurance). Note: The cost of this bond is typically 1% to 3% of the value of the shares. When the necessary information has been provided and the necessary steps are taken, a new certificate will be issued.

Stealing never transfers legal ownership. I assume what you are actually asking is whether a thief could successfully cash out stolen stock certificates. To that, the answer is not easily.

All changes in ownership of stock shares have to be registered with the transfer agent for the stock. When you present physical share certificates to a broker for sale, the broker will verify the validity of the certificate with the transfer agent. If the certificates have been reported stolen, all transfers would be blocked. Certificates do have a form for transferring ownership printed on the back, so if they hadn't been reported stolen yet, you could fill out that form and claim that you had bought or been given the certificate by the registered owner. However, you'd have to forge the signatures, and find a corrupt notary, willing to perjure themselves, to attest to the validity of the signatures.

Most other publicly traded financial instruments have similar protections. An exception would be bearer bonds. Bearer bonds are bonds for which ownership of the bond is not tracked, allowing the owner to remain anonymous. Stolen bearer bonds still wouldn't legally belong to the thief, but it would be much easier to get away with passing them off.