This behavior occurs because Queue List Views are restricted to Queue Members by default, regardless of Sharing Rules applied to the records.
If you have a Queue List View that is shared with specific Roles and non-queue member Users are receiving an "Insufficient Privileges" error when accessing the Queue, this is expected functionality.
A Sharing Rule grants access to records but does not automatically add users to the Queue. Users outside the Queue will see an 'Insufficient Privileges' error even if a Sharing Rule has been applied.
The default List View that is created in conjunction with a Queue is filtered by Queue (a radio button in the List View settings). When any List View is filtered by Queue, it is restricting visibility to Queue Members.
The Sharing Rule grants access to the records, however, it does not make them a member of the Queue.
If adding the affected users as Queue Members is not ideal, you can change the List View settings:
This returns the same contents of the Queue List View, but can be accessed by Users outside of the Queue.
Note: Possible inconsistency may be found between Classic and Lightning, where the queue list view in Lightning cases that are owned by the queue and accessible by that non-queue user is displayed.
Known Issue: Queue list view inconsistency between Classic and Lightning
000385304

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