When a user (or anonymous) doesn't have that scope, there's currently no indication whatsoever that the information on the page is out of date (on a task group for example, the running time for tasks continues to count up despite the tasks being done already since their status doesn't get refreshed) which is very confusing for people like me that don't remember that they don't have that scope unless logged in.
The only place where you can find that this is wrong is by looking at the websocket traffic in the network inspector. It would be nice to have some kind of indication on the page telling the user that they might not be receiving updates.