They are proxying through productsearch.ubuntu.com with HTTPS encryption (though there's still a a heated bug discussion over whether the search should be included at all). Unfortunately, while no personally identifying information is sent to Amazon, the privacy policy for Ubuntu sharing information about proxied searches is still unclear.

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.