A task can only be assigned to one user at a time.
You can, however, use a checklist to create subtasks and assign them to other people.
Email notifications are also sent on checklist item assignments. Also, individual users can search all boards simultaneously for their assigned work - both tasks and checklist items - so there is no risk of anyone not finding items assigned to them.
To find all your assignments, just type @XY (an @ with your initials) into the search: