My experience is it's typically a flying pilot to non-flying pilot brief about what he's going to do. The goal would be to clarify what the flying pilot is going to do and see if the non-flying pilot has any input on it.
If the non-flying pilot (NFP) doesn't know which class/category etc... non-flying pilot isn't going to provide the correct airspeed calls etc.. If NFP doesn't know you're landing on the taxiway then he's going to wonder what the hell is going on when you're not lining up for the runway etc..
Depending on the experience on type/location/operation etc... the breifing could be provided by either pilot as the other pilot might have limited experience with that scenario.
It's in our checklists for the before take-off and approach/descent checks. Of course there is on-going briefings going on between both pilots as the situation dictates.
If the company you're doing a ride with has good SOP's then basic briefings should be provided in the manuals and all you have to do is customize the briefing to your scenario.