Private hospitals should get subsidies for helping patients who cannot afford to pay for service.
Any patient in critical condition should be, by law, taken care of by the nearest private hospital. If the patient falls into the category above again some sort of tax break or subsidy would help alleviate the pressure on the hospital.
I would expect that as the pressure of taxes is taken away from private hospitals that they would be able to afford some amount of charity.
As of right now they are completely unable to function on their own because malpractice costs (and other costs) are so high with government health plans paying so little that they are being sucked up by public hospitals and becoming an administrative mess.
There shouldn't be public hospitals because then it would, selfishly I admit, raise taxes, everyone would be paying for YOUR medical bills, mostly, or at least the doctor's salary. Maybe there may be a good standard for prices in the medical field if there was a public hospital but who knows if the government would just raise it ridiculously for themselves? Maybe if there wasn't a public hospital and doctors being paid by your taxes you could have the money to go to a private hospital? As well, this would make private hospitals be more competitive and possibly lower prices.