GDP based metrics are going to assume those services are worth as much as they cost. That's definitely wrong, and everyone knows it's wrong, but you have to do something to account for it.

There are also different ways of comparing economies. The crudest is to use the exchange rates between their currencies to make the figures comparable. However, there's something called purchasing power parity (PPP) that attempts to account for differences in how much things cost locally. I don't know how PPP adjusts for government provided services, though: it's usually talked about as adjusting for market goods and services that aren't exportable or importable.

There are broader metrics that are most often used to describe developing economies. The HDI (Human Development Index) is one of the most common. It mushes together income, life expectancy, and education.