Firefox

You're probably wondering why I say "your full IP address" versus "partial IP address"; you quote the policy correctly but you missed a separate but crucial section in the privacy policy:

In addition, for security purposes and reliability of our partner’s services (detection of spam, automated activity, fraudulent clicks on advertisements …), Qwant may also collect and transfer to this partner [Microsoft Ireland] your full IP address.

The transfer happens separately from searches, sure, but if two requests get sent to Microsoft at the same time and with the same parsable information (the full IP address from the security query can be used to link a partial IP address and city-level location from a search query) then it seems like Qwant is giving Microsoft the ability, even if unintentionally, to link IP address and search.

I do not know much about DuckDuckGo, but from an initial read the privacy policy is much more vague than Qwant's, not mentioning any specific information that is shared. As they are a US company, they are also not covered by the general data protection regulation.

I agree and I'll add a disclaimer or something. DuckDuckGo says this:

In order for our product to function, we share anonymous browser and device information with our hosting and content providers for security and display purposes (for example, that you’re using a mobile device )*