Hi Caroline,
B2N2 gives wise words of advice regarding logging and claiming flight time.
I would endorse the last sentence in particular: there is nothing wrong with 'recording' unofficial RHS time in your logbook as long as you don't claim those hours towards the issue of a license or rating.
I was in a similar situation in the late 1970s. An acquaintance who was a C500 single-pilot-qualified captain used to take me along in the RHS occasionally. Over a period of two years I recorded 70 hours. When it came time for an interview for a C550 co-pilot job, the Chief Pilot was very interested that I had this experience, even though we both knew that it was unofficial and didn't count towards anything license-wise. But his viewpoint was, "you sat in the RHS and watched the operation and listened to the radio while you flew on routes similar to the ones we fly. That's useful experience in my book whether it's logged or not". I got the job!
Good luck!