Currently in Firefox's privacy browsing mode, Mozilla will provide users with the default search engine from which company they are currently using. For example, in the screenshot below, you can see that Google is used as the default search engine, and Firefox allows users to set up other search engines through access search.
To test the new features, access about:config and enable the following preferences
After that, click on the menu
If you have enabled the newly added "Show Search Suggestions in Privat"Windows"Option that displays a search recommendation when you search in Firefox's private browser mode.
When the user selects a key word on the page, the "search in private window" option pops up in the right-click menu.
Now, you can set the default search engine for Private Windows by right-clicking the options that appear when you right-click the one-time search button.