Not really from a privacy concern, that’s simply a bonus, more of a reliability/convenience interest. Even if internet/mobile internet is out for some reason, being able to easily text across local network would be nice.
For Windows/Android mainly, but crossplatform tools that include those are great too!
On Windows (NT?) you used to be able to send a message across the LAN which would just pop up in a message box. Can’t remember the command. Probably doesn’t work anymore anyway.
IIRC, it uses NetBEUI/NetBIOS, can’t realistically block it. (You could, but it’s not easy, and there’s little value in doing so).
But it’s not been included in windows since about Vista. But it still works. Just gotta get a version from Win2k.
Edit: I was thinking of Windows Chat.
Edit2: Net Send was replaced by MSG
https://briarproject.org/ It can use bluetooth, wi-fi and internet to send text messages. Also has a desktop client.