On the whole it is rare for RHS command in private flights however it might be so argued retrospectively and almost always in my experience the pilots would have had some good reason for the RHS pilot being in command if that is what they had itended.
If you find yourself in command in the RHS then make the arrangement very clear and be prepared to "carry the can" if anything goes wrong. If you are in the LHS and think you are not in command make very certain the other pilot knows he is in command and is qualified to be so.
As was previously pointed out, the OP said "aircraft", not aeroplane, so do bear in mind that helicopters are normally flown from the RHS, with one or two exceptions.
Regarding who is in charge of an aircraft or not, this report brings home how badly things can go wrong when one pilot assumes that the other might be in charge, or not.... a complete can of worms which resulted in tragedy:
http://www.aaib.gov.uk/cms_resources...BK%2012-07.pdf