I asked my friend Carmen about it. She says pensioners and people with low income do have cards that give them free care. Everyone else pays. She also reminded me that doctors recently agreed to a new contract with nearly double the pay of the old one, and that is a government-paid thing, I believe. I'm still unclear on whether that is all doctors or just those who would use the new hospital, Mater Dei, participate. I am also unclear whether doctors get one rate for hospital service and can supplement it with office visit fees like what I experienced. The reason for the big increase was that Malta was losing doctors to better paying places in the world. Now, some are returning and they are keeping a greater percentage of the newly-minted ones.
I asked if there was private medical insurance. There is, but it is generally less expensive to just pay out of hand (I am still unclear what happens in the event of a major illness for those who must pay). I asked about the low price I paid for my doctor visit, and she said the visits ranged between 2 and 4 liri. She also said it should be evaluated in light of the local income levels. But, even that doesn't explain it to an American, where the average family income for a family of 4 is $60,000 (around LM17,000)--and that's not counting the growing income inequality in the US.
SO... Malta's medical care is not socialized and may be less subsidized than I thought. But, it doesn't seem as costly as in the US for the little everyday medical troubles. I shall keep asking around and paying more attention to this question.