We don't do much with issues, but we accomplish this through task reports. The screenshot below is actually from a Dashboard, but I would think it's possible in a report showing issues. Here a writer is looking at a report, sees a task she wants to write for. She clicks once in the Assignments cell, clicks again, and is given an opportunity to select her name. Once selected, simply clicking in a different row will "activate" the choice.
My team loves this feature. They don't have to wait for me to add them to an assignment.
Hope this helps.