It really depends on what aircraft you are flying, and what skill/experience level is present in each seat. Clearly, airline and company pilots have most of this defined. For the rest of us, it is wide open.
When I am riding in right seat, I usually just ask the pilot if there is anything they would like me to do or help take care of. It might be a simple as setting transponder codes to operating the pressure control system in a more complex twin. Whatever they ask for, I do my best to get the job done. If I am left seat with a fellow pilot in the right seat, I usually ask casually what they would be comfortable with or like to do regarding the flight. I try to avoid ego based briefings or commands. A comfortable environment promotes better cooperation, and safety of flight, IMHO. It can be relaxed, while at the same time still being very professional, serious, and competent.
If I am flying with another pilot, and see something that is very wrong, I try to give help without being critical or confrontational. Don't get me wrong, with my background and experience level, if the safety of flight is in question, I will take the controls away from whomever, regardless of which seat I am occupying. But with the right communication, such action is usually unnecessary. If I really have a dispute or concern with the other pilot that might become confrontational, I save it for later on the ground. No need to get into a debate in the cockpit while there is flying to be tended to. Thoughtful discussion and potentially differences of opinion can wait.
Park you ego before getting into the cockpit. Before you think about unloading on a fellow pilot for doing something stupid, remember that you have been stupid too. We all get nervous with another pilot on board, regardless of whether they are more or less experienced than we are. The only time in my pilot life that I pulled the mixture back, instead of the prop, right after takeoff, was with an FAA inspector in the right seat. I was nervous, no question about it. The engine almost died, and I wanted to die! The inspector calmly reassured me that he was just another guy, and there was no need to be nervous. His calm action totally put me at ease, and the rest of the flight went great. If he had chewed me out, or acted incredulous that I was such an idiot, the flight would have only gotten worse. I learned a lot from his cool communication, and try to be that guy.
All for what it is worth.