At present when a role is assigned to task it shows in Primary assignment entry date field. However, when the role/team is updated by actual user who would be working on task, in Updates section we see he/she has been assigned to work on this item on so and so date.
Whereas in API we do not have field which would capture user assignment date. Also when multiple user are assigned to work on task one can mark their part as complete from notification but it is impossible at this time to know who and when their piece of work is completed at task/issue
