When a block gets broken, this message is shown to all users, even anonymous:
This block is broken or missing. You may be missing content or you might need to enable the original module.
This message should only be shown to users who have access to edit the block. However, because the block is broken we no longer know the details of the block, so it will be tricky to determine who gets to see the message. At the very least we can hide this from anonymous users.