Mmm if there were only IP microphones to connect to something like a Raspberry Pi (An Odroid H2 in my case) to have multiple mics, one on each room without the need for multiple servers...
AES67 microphones exist. Shure, Audio-Technica, and Audix are just 3 manufacturers that make them. Some of them are called Dante mics and you have to turn on AES67 in a confirmation menu. I don't know that there are any good AES67 drivers in stock Debian or Raspbian, but a lot of AES67 devices can be used with any software that supports rtsp. Whether one install of rhasspy can handle multiple streams at once is a different question. Also, even the cheapest AES67 equipment is expensive enough that a new raspberry pi or odroid h2 per room would be cheaper.