The difference is in who makes the decision, not who pays for it.
There are two ways you could have come to that decision. Let's look at them.
First of all, you might mean that in a private system everyone contributes to pay every other person's hospital bills, in which case -
No... In a private-only system all healthcare is paid for by those who need it. In a public-only system all healthcare is paid by everyone through taxes, whether through a flat tax or through proportional taxes. In a mixed system everyone pays for most things, those who want something unusual pay for it themselves on top of their general public contribution.
The second possibility is that you think money paid through taxes is private funding, in which case I imagine you consider all roads, armies and bureaucrat wages are paid through private funding, making your position utterly unworkable for any examination of private-public spheres.
So which is it?