I didn't read up on this, but essentially all web sites you interact with are
trusting multiple third parties.
This includes co-location facility, likely another middleman (company you
actually “host with”), the certificate authorities, etc. Amazon is acting
as both the co-location provider and host.
It gets more scary though. The way the system works you are actually trusting
hundreds of entities to properly verify the sites you access. If even one of
them screws up you could find that your not interacting with the true web
site that you thought you were-and/or your encrypted information is actually
being harvested by a third party.
Now think about the thousands of developers your trusting every time you use
your computer.
I'm not a fan of Amazon for so many reasons. They censor the books they sell
(physical books, they've pulled politically undesirable books, refusing to
sell them, but technically legal), they violate users rights by pulling
content sold, they push digital restrictions on users, and much more.
That said I'm not sure that DuckDuckGo's use of Amazon (it sounds like it is
though) is risking users privacy. It's possible DuckDuckGo's using Amazon in
ways that it doesn't. My understanding is if your going to compete with
Google, Bing, and ???, then you need significant infrastructure to make it
happen. I doubt DuckDuckGo has the financial resources to build its own data
centers and maintain them. They might not be able to afford that, but they
may be able to afford to rent a data center from Amazon if Amazon will let
them rent it by the hour. Now all of a sudden it becomes cost effective and
feasible to compete with Google, Bing, etc.
I would fight tooth and nail to avoid Amazon, but I don't know what other
options exist. Unfortunately in many cases there are no good solutions.