Process Assignments web part (Current Assignments in ServiceDesk's default SD Incident Page) does not show the assignments for the process without the necessary permissions set.

The permission required for Process Assignments web part is:
WorkflowTasksManagement.ViewUnassignedTasks

This permission is specific to the web part, seeing the Process View Page at all still has the usual requirements of at least Can View permission on the process as well as permission to the desired Process View Page.