I believe in freedom to spend you money how you wish, so wouldn't get rid of a private system altogether, but think that if we are going to have a two tier system then they should be entirely separate. A private system should train its own staff, build its own hospitals and not rely on the NHS (which is paid for with public money) at all.
I particularly dislike the system under which consultants use NHS clinics and hospitals to see private patients, who can 'jump the queue' by paying for an initial consultation. To me, that is really wrong, although if a patient is desperate (eg at risk of long-term damage if they stay in the queue), I can't honestly blame them for doing it, so long as they don't claim to be 'freeing up a space' for an NHS patient. They are absolutely not doing that, they are pushing ahead.