The trade-offs are fairly consistent worldwide. Public systems, where you can join them, are usually cheaper, income-based rather than risk-based, cover pre-existing conditions without underwriting, and give you the same entitlements as citizens. Their weaknesses are waiting times for non-urgent specialists and elective procedures, less choice of doctor, facilities that vary in comfort, and, relevant for expats, limited English or foreign-language support outside big cities.
Private insurance buys speed and choice: faster specialist appointments, private hospitals with more comfort, often English-speaking staff, and direct access without gatekeeping referrals. The costs are premiums that rise with age and claims, medical underwriting that can exclude pre-existing conditions, and coverage caps and exclusions that need actual reading.
Many expats land on a hybrid: join the public system for the safety net of major and chronic care, and add modest private cover or simply pay out of pocket for quick outpatient visits. Which balance makes sense depends heavily on the country, since the quality gap between public and private ranges from negligible to enormous.