I think it depends on a lot of things and the right answer is something that might be different for each family, but needs a proper conversation so everyone understands and accepts the reasoning for the approach used.
When I lived with my parents, I was lucky enough that they didn't charge me board. I offered to but after a discussion with my dad he basically said that he flat-out didn't need the money and the bill increase with me there was negligible. He had no use for my money so he wanted me to keep it.
I did, however, pitch in with all the household chores including cooking for the family a couple of nights a week and would pay for the big shop once every few weeks too, as well as helping my parents out with my "young person expertise" when they had IT needs and doing my fair share of manual labour when they were doing home improvements etc.
I worked full-time and was able to save up a good deposit for a house, and the only assistance my parents gave to me when I moved was advice and helping out with the heavy lifting, whereas many of my friends who paid board had their parents contribute to their deposits.
Although I don't come from a particularly wealthy background, I appreciate I was in a very privileged position there, both financially and in terms of my relationship with my family.
