If you click the Chat button on a DDG search page, it says:
DuckDuckGo AI Chat is a private AI-powered chat service that currently supports OpenAI’s GPT-3.5 and Anthropic’s Claude chat models.
So at minimum they are sharing data with one additional third party, either OpenAI or Anthropic depending on which model you choose.
OpenAI and Anthropic have similar terms and conditions for enterprise customers. They are not completely transparent and any given enterprise could have their own custom license terms, but my understanding is that they generally will not store queries or use them for training purposes. You'd better seek clarification from DDG. I was not able to find information on this in DDG's privacy policy.
Obviously, this is not legal advice, and I do not speak for any of these companies. This is just my understanding based on the last time I looked over the OpenAI and Anthropic privacy policies, which was a few months ago.