When your users contact you saying they cannot share a case or an activity with another user - the first thing to do is to check the basic security role settings for Cases, Appointments and related entities (such as Leads, Opportunities, Contacts - depending on what the Appointments are related to). If those settings all appear to be correct (I won't try to document the correct settings here), then you should also check your Queue settings.
Yes, your queue settings. On the Core Records tab.
Make sure that you have the security role set to read for the Organization. See the image below.
You are probably asking, "Why the Queue entity?" Activities and cases are both listed in your queues. So if you haven't granted permission to read the Queue, then you can't assign your activities or cases to another person's queue.
You may now be asking, "We don't use Queue's. How can I get rid of the Queue item from the nav bar using Security Roles while still allowing users to reassign activities and cases?" That's an easy one: You Can't. But you can update the site map.