We're very open with our son, who is now 11. We involve him in planning for the family's major purchases. A couple months ago we purchased a house as an investment, and he was very involved in that decision-making, as he is now in charge of yard work and snow removal at that property. He budgets his own money and does a very good job saving, sometimes for short-term goals, but more often for college.
I take care of most of the bill-paying and budgeting, but on the first of every month I enter our information in a spreadsheet and all three of us (son, husband, and I) sit down and look at it. I think it helps keep everyone on the same page. Before we did that, my husband was never exactly sure what we could or couldn't afford at any given time.
My parents never shared the family budget with me when I was growing up. I only knew "okay" or "we can't afford it." I think if I had known a little more, maybe I would have made different choices about what I asked them for. (Field trips, music lessons, summer camp...)