Thanks for raising this question around assignees marking tasks as complete. I'm not sure which version of Jive you're on, but in later versions, the assignee should be able to mark it as completed by placing a checkmark on the task. On the opposite end, if you need to undo this, click on the Archived link and remove the checkmark from previously completed tasks to flip it back to in progress (or overdue, if past the due date).
I hope this helps. Let me know if you need anything else.