In most developing and middle-income countries, private hospitals are indeed where expats and even affluent locals go, and the reasons are practical: shorter waits, English-speaking staff, better amenities, and smoother paperwork for insurance claims. Public hospitals often have excellent doctors, sometimes the very same ones moonlighting privately, but they can be crowded, under-resourced and challenging without local language skills. The main exception is major trauma and emergencies, where the biggest public hospital is sometimes the best-equipped place in the region.
Costs without insurance vary enormously by country, from remarkably affordable consultations in South and Southeast Asia to genuinely ruinous bills in the United States, so it is impossible and unwise to quote figures generally. What is universal is that private hospitals usually want payment or a deposit upfront from uninsured foreigners, and costs escalate fast for surgery, intensive care and imaging.
The practical takeaway: never travel or move abroad without health coverage, whether travel insurance, an expat health plan, or enrollment in the local system where your visa allows it. Ask local expat groups which hospitals they actually use; that word of mouth is more current than any ranking.