how does the healthcare system works in Myanmar ? Is it efficient ?
=> Oh no 
What are the main differences between public and private sectors?
=> I would recommend using °µÍø½ûÇø doctors and clinics, and the local doctors recommended by the °µÍø½ûÇøs (e.g. SOS Clinic, Australian Embassy Clinic in Yangon). Public hospitals are in a miserable condition, some private hospitals are well-equipped (e.g. Pan Hlaing Hospital in Yangon) but don't usually have enough doctors based at the hospital.
Is it recommended to purchase private health insurance in Myanmar?
=> Never heard of such a thing in Myanmar! Cash is what you need, and get a good international cover insurance back home