Every family's financial circumstances are different, but there should be no doubt whatsoever that a huge part of 'growing up' is taking financial responsibility for yourself, whether you live at home or independently.
I'd believe I'd failed as a parent if a child didn't voluntarily offer to pay towards their upkeep after school leaving age. It would imply they had no sense of value or fairness. And I'd also be failing as a parent if I took all they offered and spent it... unless, of course, the family was struggling.
This basically sums it up.