A Plex support page states that "Plex Web App does not yet support shared syncing". It suggest that you instead, "use the device to which you're syncing"; however, as another feature request mentions - the desktop clients do not support Sync either.
Each member of my family has a myPlex account and I shared select folders with them. I would like them to be able to sync movies from their shared folders to their laptops for plane trips etc.