We use Sub-tasks & Defects (a Sub-task Issue Type) to assign work instead of assigning people to a Story which is really owned by the Team.
We organize our Agile Board Work View with Swimlanes for Epics and use the prioritization of the Epics to determine priority within the currently running Sprint.
Epics contain Stories which contain Sub-tasks & Defects which are then assigned to users.

However, during our Daily Stand Up, when we use a filter to see what work someone has the filter hides the Stories (as we intend) to which the Sub-tasks & Defects belong, but all of the Sub-tasks & Defects then show in the "Issues without Epics" Swimlane instead of the Swimlane to which their containing Story is a part.