Both models exist, and knowing which you're facing saves real stress. Direct billing, where the hospital settles with your insurer and you pay only your excess, is common at large private and international hospitals that have agreements with major insurers or work with assistance companies. It usually requires activating it: you or the hospital's international patient office calls your insurer's assistance line, the insurer verifies cover and issues a guarantee of payment, and treatment proceeds. This is why the assistance number, not just the policy paper, is the thing to keep handy.
Smaller clinics, public hospitals, and outpatient visits more often run pay-and-claim: you settle the bill yourself and submit for reimbursement. That's workable if you're prepared, so carry a credit card with headroom, and collect everything, itemized bills, medical reports, prescriptions, receipts, since claims fail on missing paperwork far more than on coverage disputes.