It's quite simple. DuckDuckStart doesn't track any queries submitted whatsoever, It simply redirects you to DuckDuckGo or Startpage depending on whether your search contains a !bang or not.
The source code for this is available here on GitHub so you're free to take a look or even host your own version if you'd like.
The random part of the URL that appears when you search on Startpage sometimes is to prevent links clicked on in the results from leaking your search query. Startpage will leak your search query if it is not submitted via a POST request. This will only affect your queries if you add DuckDuckStart to your browser.