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.
We do use Plausible Analytics to measure traffic as well. They are a privacy focused web analytics solution. You can view their policies here.