Plex definitely connects to third party servers, primarily for verifying subscription status (some features are tied to a subscription) and to make connecting to your Plex box from outside of your local network easier (no IP addresses or something like dyndns).
Kodi I’m not sure about. It was originally XBMC (XBox Media Center) and is open source so even if there’s so phone home element, it can be built without that. Another open source option is Jellyfin, which is a fork of Emby from before that project was closed off.
Why does plex have subscriptions at all? I understand (at least theoretically) that not all software is free, but charging for it the normal way (you pay money and buy a copy that you then use without it phoning home) works perfectly well too. Anyway I had had the impression that Plex also sends at least the metadata of your video library to its home server.
I didn't realize Kodi was XBMC or that you could build it yourself. Thanks for that info.
Kodi I’m not sure about. It was originally XBMC (XBox Media Center) and is open source so even if there’s so phone home element, it can be built without that. Another open source option is Jellyfin, which is a fork of Emby from before that project was closed off.