If we look at a country level then things like attitude towards education, the specific implementation differing between countries, or the general social structure outside of school differing may make comparisons meaningless.

Another factor is that “charter school” doesn’t necessarily mean run by a company, just that they have an agreement with the government (a charter) that says they don’t have to follow the normal curriculum. But from my understanding charter schools in NZ are privately run?

Regardless, there are some studies. Here’s an article about one:

The figure below shows some notable results from the CREDO studies. The key takeaway is that charter school students, in general, perform about the same as their matched peers in the traditional public schools, but there is variation across different types of schools and groups of students. For example, students in urban charter schools generally perform better than their matched pairs—likely for an assortment of reasons—while students in online charter schools perform much worse.

This study shows that charter schools are slow to get started (i.e. perform poorly at the start) and slowly catch up to public schools over time:

This study investigates whether student achievement varies during the institutional life span of charter schools by comparing them to new public schools. The results show that there is little evidence that new public schools struggle with initial start-up issues to the same extent as new charter schools. Even after controlling for school characteristics, new public schools generally perform about as well as one would predict given their demographic and socioeconomic profile. New public schools hit the ground running and maintain steady performance, while new charter schools begin to improve after their first year and slowly close the gap.

This article talks about how charter schools and public schools are as good as the people running them:

The effectiveness of charter schools is a hotly contested and often debated issue among educators, parents, researchers and politicians. And studies have been published supporting both sides of the argument: that charters underperform public schools and that they outperform public schools. However, the key difference appears to be the state in which the charter is located and the organizing body by which the charter is run.