zany500, in Hong Kong you can't choose your doctor if you are in the public system nor can you choose the hospital you attend. The hospital that you will be sent to is the one which you are geographically zoned to and once in that hospital system you see whoever gets assigned your file that day.
Of course there are exceptions to this rule e.g. if you need to have a liver transplant or some other super specialised care, you will be funneled to the one centre that takes care of that condition.
If however you are asking for public paediatricians to see privately, then that is a totally different ball game and yes you can choose but you will have to pay for the consultations and if you end up in hospital you MUST be admitted privately into the private part of the public hospital.
Hope this has helped.