It is understood that a user in unable to assign a task from a public folder.
However, I would like to know what (if any) Microsoft applications, add-ins,
create ideas any may know about regarding the following scenario:

From a shared/centralized location, multiple users must have the ability to
create and assign a "task" to another person (in another department); the
assignee updates the status, and defers to another user, etc. until the
process is complete and the task is marked accordingly. All users must be
able to see the current status as needed at any given time (allowing that the
task is being updated).

Your suggestions are appreciated and/or if you know of other means to solve
this much needed streamlining process.