Return to past conversations with Duck.ai’s Recent Chats feature, which stores chats locally on your device – not on DuckDuckGo or other remote servers. Recent chats can be deleted individually or with a click of the Fire Button. You can add or remove the Duck.ai button in your DuckDuckGo browser from browser settings, and hide the Duck.ai buttons on the search results page from search settings.
From the DDG Help Page
Chats are stored in your browser cache