As for Plex’ dependency on the internet you can read up in this article from the tips & tricks section
If your server is not visible from within your home network without remote access being active, that implies you have a problem in your network setup (obviously). Check if the devices are indeed on the same network. Some routers/access points create their own subnets instead of expanding the network they’re added to. With a Plex Pass you can also configure Plex to recognize other subnets to be part of your LAN (see Settings > [Server Name] > Network > LAN networks)
Make sure your home network is using an actual private network IP address range.
