You know,
it changes from class to class, depending on class need. The key is, whatever grade level you're using it with, that you're using it as a tool to build a connection and a relationship with them. It's all part of getting them on your side, and when they're on your side, they will lead the class in the direction you want them to go.
Some examples of my 7th graders' leader responsibilities and privileges this past year:
Attend weekly leadership meetings, where I usually provide some sort of treat, lunch, etc... they loved that.
Assign each leader to identify a student that needs a 'nudge' to get on board, and secretly 'take them under their wing' to steer them in the right direction.
Choose a student to receive a special leadership award each week based on their observations.
Plan videos to demonstrate WBT techniques (they REALLY wanted to get on the site)
Any helpful task that lets them get out of their seat... pass something out, get something down for me, hang something up, walk a message down to the office, escort another student, etc.... middle schoolers will do ANYTHING to get out of that seat.
Controlling timers for various activities
'Scoring' the classroom on progress toward certain procedure-related or behavior-related goals
"Table leaders" if you have them set up in tables....especially if there's a competition. The winning table simply got to eat during class that Friday. I didn't provide the treat, I just said they could eat....but only the winning table (got this idea from Angela Macias!)
Their biggest responsibility is to put pressure on others to 'get it together'...whether by a look, a nudge, a whisper, help encourage their classmates to stay within our game.
They are evaluated by their peers every week with official 'elections' every two weeks. This reminds them that if they don't step up, they will lose this 'status' and privilege.
Remember, I had a *horribly behaved* group of middle schoolers, and they still ate this up. Leaders were often the popular kids who used to be a behavior problem. those are often your best leaders because they already are peer leaders.
The key to the leadership system: Awards!! Find many awards that you can give weekly or bi weekly to recognize various aspects of leadership in the classroom to as many students as you can. If I missed a week, they knew it, and they mentioned it. They really looked forward to those award certificates.