Vivaldis has collaborated with DuckDuckGo to use its search engine as default in Private Windows and is the first browser to do so.
Vivaldi 1.14 stable update now available with the Private Search feature. Vivaldi browser uses DuckDuckGo as default search engine in Private Windows. This update introduces private search feature and allows to set any search engine you like as default in Private browsing mode.

DuckDuckGo is the default Private Search engine in Vivaldi
Do know, the private Search engine feature was already available in the snapshot version of Vivaldi 1.15.
READ: Vivaldi 1.15: Default Search Engine for Private Windows
The Vivaldi team says in their blog, with this change, searching in private window will be more really private and believes user search history won’t be tracked by DuckDuckGo.
To experience Private search in Vivaldi with DuckDuckGo engine
Click on Vivladi V icon > File > New Private Window
Private Window introduction page that appears (which you can set to not show again) allows to choose and set default search in Private Windows from Bing, Yahoo, StartPage, Wikipedia, Ecosia and Google.

And you can anytime visit Vivaldi Search Settings and make desired search engine as default in regular and private windows also.
Also read: How to Change Vivaldi browser’s Default Search Engine to Google