It seems like all user input to the Dash home lens is send to Amazon by default in Ubuntu 12.10, is this correct?
Also, there are concerns that the image thumbnails of results that users download could allow Amazon to correlate queries to IP addresses.
Shuttleworth also wrote a blog post that tries to address the privacy issues with this feature.
It is not unreasonable to think that Canonical keeps logs on all of its servers for some period of time and maybe even backups.
Dash search terms are not sent to Amazon directly. They're sent to Canonical, which then sends them to Amazon. That means Amazon doesn't get your IP.
However, as of the current implementation, thumbnail requests are directly sent to Amazon. So Amazon does get your IP in the end.
That is correct. Dash in Unity leaks information to Amazon and others.
You can read more about the privacy issues and how to fix them in EFF's post Privacy in Ubuntu 12.10: Amazon Ads and Data Leaks.