Usually a USB mixer is just an analogue (conventional) mixing desk, with Analogue to Digital converters built in, and a USB interface added so that the digital data can be sent to your computer. Some USB mixers have just a stereo A/D converter allowing you to transfer the main stereo mix to the computer, while others have multiple converters that allow multiple individual channels to be sent to the computer.
Audacity does not “communicate” with the mixing desk other than receiving audio data from it, and (possibly) sending audio data back to it.
For interaction between the mixing desk and the computer software, you need something like a Digidesign set-up (Pro-Tools).