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.
Here is a snippet of our logs. We don't save any queries.
127.0.0.2 - - [22/Jun/2018:16:56:25 +0000] "GET / HTTP/2.0" 200 2772 "" ""127.0.0.2 - - [22/Jun/2018:16:56:25 +0000] "GET /js/material.min.js HTTP/2.0" 200 11582 "" ""127.0.0.2 - - [22/Jun/2018:16:56:28 +0000] "GET /favicon.ico HTTP/2.0" 200 773 "" ""127.0.0.2 - - [22/Jun/2018:16:56:28 +0000] "GET /opensearch.xml HTTP/2.0" 200 583 "" ""::2 - - [22/Jun/2018:17:05:21 +0000] "POST / HTTP/2.0" 302 188 "" ""127.0.0.2 - - [22/Jun/2018:17:06:37 +0000] "GET /?query=" 307 173 "" ""127.0.0.2 - - [22/Jun/2018:17:15:27 +0000] "GET /?query=" 307 173 "" ""127.0.0.2 - - [22/Jun/2018:17:15:36 +0000] "GET /?query=" 307 177 "" ""127.0.0.2 - - [22/Jun/2018:17:16:10 +0000] "POST / HTTP/2.0" 302 142 "" ""127.0.0.2 - - [22/Jun/2018:17:17:43 +0000] "GET / HTTP/2.0" 200 2772 "" ""127.0.0.2 - - [22/Jun/2018:17:17:45 +0000] "GET /js/material.min.js HTTP/2.0" 200 11582 "" ""127.0.0.2 - - [22/Jun/2018:17:17:47 +0000] "GET /opensearch.xml HTTP/2.0" 200 583 "" ""127.0.0.2 - - [22/Jun/2018:17:17:48 +0000] "GET /favicon.ico HTTP/2.0" 200 773 "" ""127.0.0.2 - - [22/Jun/2018:17:22:13 +0000] "POST / HTTP/2.0" 307 127 "" ""127.0.0.2 - - [22/Jun/2018:17:22:34 +0000] "GET / HTTP/1.1" 200 7252 "" ""127.0.0.2 - - [22/Jun/2018:17:23:04 +0000] "GET / HTTP/1.1" 200 7252 "" "Bot"127.0.0.2 - - [22/Jun/2018:17:24:12 +0000] "GET /?query=" 307 203 "" ""
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.
When using DuckDuckGo you will see your search query in the URL. This is okay. When you click on a result DuckDuckGo will redirect you through their site to prevent your "referers" from leaking your search query. DuckDuckGo explains their redirect.