I just checked, and Plex happily played my local media with my internet connection powered down. I'm using Plex 1.16.5.1554 running in a jail via the FreeNAS plugin and playing from a Roku Stick+.
Only hiccup was the Roku client took unusually long to bring up my list of local content; I could imagine it timing out on DNS connections.
I've never connected it to the Plex servers ("associated it with my Plex account"), perhaps thats the difference?