In our system, based on imported data, we have the role of Pastor (whose parent is individual), and pastors are the only contacts with Drupal accounts.
When a pastor logs in, they are shown the dashboard for their Household, but this is, of course, empty, as all cases and activities have been assigned to them as an individual.
How can we correct this?
There are multiple things called dashboard, but if I understand right I think the one you're looking for is the "Case" dashboard, which is not the default dashboard unless you change the frontpage under /admin/config/system/site-information. The case dashboard is under the Cases menu.
In this instance what wasn't working was that the Drupal account had been created manually rather than from CiviCRM, and therefore there was no link between the Civi contact and the user account. Interestingly there doesn't seem to be any UI for linking an existing contact to an existing Drupal account. (I'm happy to be corrected on this!) Research gave me the options of directly adding a record to the database or of deleting the Drupal account and recreating it from the contact. Since the account owned no content, I chose the latter.
I don't seem to be able to comment so adding another answer to address the UI for linking. CiviCRM will link automatically upon login, but the rules depend on the email address and the dedupe rules you have configured on your site (under Contacts - Find and Merge Duplicates). See drupal/civicrm_user.inc where the login hook is and CRM_Core_BAO_UFMatch::synchronize(). See also http://wiki.civicrm.org/confluence/pages/viewpage.action?pageId=86213711
