It depends on the financial / professional situation of each parent and custodial time, but even if it's 50/50 time and similar incomes, you may be required to pay support. Even if one parent is rich and the other parent is very rich, the very rich parent will likely pay child support to the rich parent.
You should contact an attorney, your ex will. You need to find out if you will be paying for private school on top of child support, unless you don't mind. My son's private schools have gone up 5 - 10% a year, higher a few years as private schools do. If your child is in a reasonably priced K - 8, what about high school, tuition can easily double. What about uncovered medical expenses (high deductible policies), are they going to be split 50/50? Has she already had braces? Maybe you have orthodontic insurance now, but we've had it change year to year and we always have great insurance, depends on your employer. (Seems like every kid has braces now, especially in private school. $6500+ in L.A.)
No matter how friendly things are right now, they are likely to become unfriendly. = ( Sometimes when parents find out that the amount of custodial time can determine child support, that starts the war.