The answer in short is yes. I was diagnosed with a Chronic condition in HK. I went private to get diagnoses (I originally went to the Hospital Authority but was given an appointment a year later) and then like you found the drugs very expensive.
However, having received the diagnoses privately I went back to the gov. system and got an appointment much quicker. I then received my meds from the gov hospital and regular specialist appointments there. I occasionally went back to the private specialist too. They have a system where the HA can share your records with the private clinic using a code.
Of course not all drugs are available for free from the HA. I 'made do' with something alternative. (Which, interestingly, the UK doctor took me off as soon as I went back to the UK. Dangerous and not used in the UK for years he said!)
Do be aware though. I found the private specialist was trying to push me on even more expensive drugs (Biologics) which in practice were not needed.In time I realised that my condition was not as severe as it could be and I am able to get by on minimal, cheaper meds.
I wonder if private specialists get a kick back from prescribing certain expensive drugs.