Considering that most private clinics are rather smaller than hospitals, that there are only a handful of hospital-sized private establishments in Greece (mostly in major cities, and often specialising on maternity or psychiatric care) and that most hospitals in the country are public, I'd expect most people who use the word
νοσοκομείο to have a public hospital in mind, especially in case of an emergency.
I don't think there's a strict definition, however. The Henry Dunant Hospital Center in Athens is also known as a hospital in Greek, and it's always been private; others started off as private foundations and were later integrated into the National Health System, such as "Evangelismos" (also in Athens). Judging from the title of
this list, and the fact that the word
νοσοκομείο doesn't really appear as such in the legal titles of the businesses named there – even for establishments with hundreds of beds – it is possible that a legal distinction exists. We do find
νοσοκομείο in two cases of charitable foundations in the islands of
Aegina and
Hydra, but they are very small; historically they may be entitled to the name, but I doubt people would refer to them as "hospitals" in everyday speech.
Someone who lives in a place where people actually visit private hospitals might have more to say, but I think it's generally a matter of size. It rather reminds me of
how the word city is used in the United Kingdom, where official and everyday use differs.