User Agent: Mozilla/5.0 (X11; Linux x86_64; rv:60.0) Gecko/20100101 Firefox/60.0
Steps to reproduce:
Opened Edit -> Preferences -> Search, or about:preferences#search
I can configure web search engines there.
I think there should be "Private Browsing" checkboxes for both the "Default Search Engine" and the "One-Click Search Engines" section, used to enable private browsing for the web search.