The toolbar is not just for administrators. Previously, an empty 'Manage' tray would always appear, even for users without permission to see any links. Now, to view the 'Manage' tray, a user must belong to a role with at least one of these administration permissions:
access administration pagesaccess content overviewaccess media overviewadminister blocksadminister commentsadminister nodes
Due to other limitations and assumptions, existing sites probably already grant one of those permissions to any user configured to see the toolbar. However, it's possible that some roles with elevated permissions providing access to links inside the "Manage" tray of the toolbar did not also include the access administration pages permission or any others from this list. Users in these roles will not be able to view the "Manage" tray until their role(s) grant at least one of these permissions.