Unable edit a coworkers Tasks and Events

The ability to view and edit activities in Salesforce is based on the role hierarchy and Sharing Settings.

Based on role hierarchy: 
Someone who is higher in the role hierarchy can edit tasks and events assigned to the users who report to them in the hierarchy, but subordinates cannot edit activities assigned to someone higher in the hierarchy or activities assigned to users with the same role.

Based on sharing settings:
If your organization has set Activities to Controlled by Parent in your sharing settings, then users can edit Tasks and Events owned by other users provided they are able to edit the parent records associated with the Activity.

All users are able to edit, close, and delete the tasks that have been assigned to them.


See also:
Cannot view or edit an Event from a shared Calendar