>As it stands, they should append this to the end:
>"but the architecture of our site lets Amazon log it."
This is the best suggestion yet. Of course there are numerous hacks possible to make it more secure, but each one at the cost of user experience. DDG has so many things going for it, I don't think for the majority of users security is at the top of their list.
Do a http redirect to a url which has an encrypted version of the search term in the url parameters before displaying the search results. That fixes the problem without hitting the user experience.
However, at the end of the day, everything can be logged anyways. If someone is really concerned about this, they shouldn't be using the internet.
Edit: I want to be clear. If someone is concerned about the fact that things can be logged, then they are better off not using the internet, as pretty much everything has the potential to be logged. This is, after all, what the complaint is: the potential of being logged.
"We don't log your IP address with the search term"
As it stands, they should append this to the end:
"but the architecture of our site lets Amazon log it."