Healthcare in Slovakia is efficient in general, especially in bigger cities. Doctors are reliable and they're keeping up-to-date. Especially those in the public sector might be a bit overworked, since they usually take care about more patients than is the average in the western Europe - don't be surprised to meet a doctor in not-the-best-mood, however they still do a good job. Public insurance covers any potential healthcare you would need I guess. It is usually possible to pay extra for better comfort - e.g , better room in the hospital, setting exact time for visiting a doctor (normally you have to wait in the queue), extra vaccination, etc. This payment is made directly to the given doctor/clinic/hospital. Private clinics offer better comfort for an annual fee (starting from 250 eur in Bratislava), however they are still based on the public insurance system. They also offer services for people without public insurance, but I don't know about fees. Important for foreigners: their doctors usually speak several languages other than slovak. These private clinics do not cover complicated treatment and surgeries, that is covered only by large state clinics/hospitals.
Due to the system of direct payments for extra services mentined above, private health insurance is used rarely in Slovakia.