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Are you using a particular project management methodology (e.g. Scrum)? And are you part of an agency or other business that has clients? In the latter case I think timesheets are pretty common so you have a paper trail for billing purposes.
But don't assume that a PM will always take responsibility for assigning issues to team members. In Kanban (or at least in many implementations of Kanban) teams are often self-organising in the sense that a backlog will be provided and team members are free to work on whatever's top of that list once they become available. In effect, they're assigining issues to themselves.
