What triggers the "would like to find and connect to devices on your local network" permission notification on iOS 14?

In a nutshell, Bonjour. Its use is no longer "transparent". See https://developer.apple.com/videos/play/wwdc2020/10110/ for more information:

If your app interacts with devices using Bonjour or other local networking protocols, you must add support for local network privacy permissions in iOS 14.

Even an existing app is subject to this rule; the first attempt to use Bonjour triggers the authorization alert.