I just spent an hour writing a great article about the Chinese government's moves to encourage public hospital doctors to work in private practice. But my internet browser crashed and I lost it.
Short version: there is little enthusiasm for this move by doctors because of the practical barriers: pensions, medical indemnity insurance and new contracts on working hours/remuneration. Also barriers because public hospitals don't want to lose their 'talent' and see the same doctors working in 'competition' against them (not to mention the conflict of interest this creates). And importantly for doctors, their careers are determined by the public hospital system, from which they get their experience, evaluations, academic training and status.
Conclusion: Moving to more flexible public/private working arrangements in healthcare is an immense challenge for China and won't happen overnight.