NOTE: This is an unofficial answer by a forum member like you. :) Please contact Amazon for official answers.
Many Kindle owners become very fond of the devices. It's devastating for many to think of their Kindles being lost or stolen, but unfortunately, it does happen. Kindles have a high value on the secondary market (E-Bay, Craig's List, local newspapers), so there is definitely motivation for thieves.
If it does happen, what should you do?
The first question is whether there is a risk that someone has it that you don't trust. You may feel different if you left it at a family member's house than if you left it in a restaurant.
If a "bad guy" could have it, deregister the device. That is what Amazon recommends here:
If your Kindle is lost or stolen, or you transfer ownership to another person, you will need to deregister your Kindle from your Amazon.com account.
While registered, your Kindle can access your account. A user can make purchases using your 1-click settings, which could place significant charges on your credit card. If you have a gift card/certificate balance on your account, your 1-click purchases will first draw from that, before using the credit card.
It's important to note that the Amazon policy says:
We are not responsible if a Gift Card is lost, stolen, destroyed or used without your permission.
According to the policy, then, if your Kindle is stolen and you have a gift card/certificate balance and purchases are made, you will be out that money.
If you deregister the Kindle, it can not access the Whispernet (the Kindle's internet access), and it can not be used for purchases.
If you do not deregister it, the finder can deregister it directly from the Kindle, so you do not protect yourself by leaving it registered.
EDIT: Maggie Leung, a Kindle forum member, also pointed out that a thief could use a stolen Kindle to get into your e-mail accounts or other websites if your password is stored. This could let them get to confidential information. Deregistering will prevent this.
What else should you do besides deregistering it?
1. Complete a police report. Although a Kindle is likely to be considered petty theft (the typical cut-off is $400: while a Kindle DX costs more than that, it may not be assessed at new value). The police may be able to recover it, and having a police report may help with other steps
2. It's possible your renter's/homeowner's insurance may come into play. Check your deductible: it may be higher than the value of the Kindle. However, if the Kindle was only one of the things stolen, you may have a claim
3. You may want to check the Lost & Found where you were. Some people have reported success with that
4. Contact Amazon, and ask them to "blacklist" the device. When it is blacklisted, it can not be registered by someone else. If it does show up, you can call them and they can unblacklist it so you can re-register it
5. Edited to add: if you have any subscriptions through the Kindle store (magazines, newspapers, blogs) which are delivered to the lost Kindle first, go to http://www.amazon.com/manageyourkindle, and click or tap "Subscription Settings". You can cancel them or have them sent to another device. Deregistering a Kindle will not stop you from being charged for a subscription going to that device
Amazon is not an enforcement agency, and will not be the ones going after a thief. They do not have a way to verify your report that it was stolen. You could have sold it to someone else, and then reported it stolen. Another likely scenario is that a thief steals it from Customer A and sells it to innocent Customer B. Amazon can not compromise Customer B's privacy by giving any information about that person to Customer A.
If you report it to the police, it's reasonable that Amazon would cooperate with an investigation. In the scenario above, when Customer B went to register it, Amazon could report it to the police (if the police had made such a request). The police might then recover it, and return it to Customer A (after any ownership dispute was resolved). Catching the thief might be more resource-intensive, and Customer B would probably be out both the money and the Kindle.
If your Kindle is recovered and you reregister it, you will still have access to your Kindle store items. If you had personal documents or books from other sources, they will still be on the Kindle unless they were deleted by a thief (who might just do a factory reset).
If you get a new Kindle, you will have access to books you previously bought from the Kindle store (but not back issues of subscriptions). See:
If you buy a new Kindle, and the old one is recovered within thirty days, you can return the new Kindle for a full refund (you need the original packaging and accessories and it needs to be in original condition). The instructions are here:
NOTE: You may want to consider a recovery service like TrackItBack or Stuffbak. You purchase a sticker, which has a phone number a finder can call. The recovery service acts as a go-between, and no personal information is exchanged.
For more Frequently Asked Kindle Questions, please see:
(Amazon threads and pages relevant to the discussion)