Currently, a user needs to belong to a Group with at least the Privileged User Role, and needs Share access to the dashboard, in order to be able to schedule the delivery of that dashboard via email. This opens up a scenario where a Finance user could accidentally or maliciously share access to a dashboard with a Marketing Group, potentially exposing sensitive data.
My gut feeling would be to have a setting within each dashboard that enables scheduling for a set of Groups. Adding a new Schedule access right to the existing View, Share, Edit access options on a dashboard could also be another approach. This should remove the need to have Share access to the dashboard so the Finance and Marketing scenario above is not possible.