I'm a huge Amazon shopper and have been a Prime member for the last year or so and love both amazon and prime. I think this is a pretty innovative idea.
I'm not really falling into the fears that others are having though. I've seen the drones my LHS builds and the GPS on these units are amazing and extremely accurate. I'm sure as time goes forward (2015 or later) that GPS is only going to improve on our "toy" drones.
That said I'm sure what ever Amazon uses will be as good and likely better than our toy drones. So the idea of this dropping in the road, river, lake doesn't seem very likely.
As far as theft, this would be no different then the UPS, FedEx, or USPS leaving your package at the door. In fact I would believe theft would be less of an issue as those that would opt for the 30 minute delivery are likely doing so because they need it quick and either are already at said location or will be very soon. Actually this could be more secure as you could have them delivery in your back yard and not in the front for all to see.
Really the issue I see is there are places that encourage or in the process of encouraging places to shoot down drones. The simple fix for that is for Amazon to not allow 30 minute delivery in those areas - problem solved.
This would be great for a number of reasons, even RC. Imagine you're out at the track and the LHS is closed or not close by. You break a part, need a new set of batteries, maybe a tool, etc. Instead of calling it a day order it from Amazon and have it in 30 minutes. Very cool!
Check out 60 minutes interview,
http://www.cbsnews.com/news/amazons-jeff-bezos-looks-to-the-future/ - talk of the drones start at 11:15