As true as that may be, it's not entirely accurate. Although you do rub shoulders, their education system is much better than most public schools for a few reasons:

• Very low student-to-teacher ratios
• Much higher qualified teachers
• A lot more homework (I have friends who went to a private school that worked harder there than I currently do in my aerospace engineering bachelor's)
• They teach you real life lessons and don't have a strict curriculum to follow
• They ensure each student is involved in some sort of club or sport that they enjoy

...there's probably more to that list. The biggest difference is that they don't have to just teach you that standard crap that prepares you for nothing. You get one-on-one attention from very high qualified teachers that give a shit about you actually succeeding in life. It's a steep price to pay, but I can completely understand why you would if its within your means.

EDIT: I went to a public school and absolutely loved it. I had great teachers. On AVERAGE, you're more likely to get a better education at a private school, just like you're more likely to get a better education at Harvard than you are at Carleton (that's my university in Ottawa, Canada). It doesn't mean that I won't encounter great professors.