I prefer a single queue with simple rules of who can do what.

The approach I learned at Pivotal Labs was that product management decides the order of stories and bugs, since these add or subtract user value. Engineering can put chores anywhere in the backlog according to their considered discretion, since these reduce drag and risk. Each week you have the Iteration Planning Meeting to look at what's coming up. Each day engineering pulls whatever's on the top of the backlog and works on it until it's done.

It seems really straightforward to me.