I’ve seen so many people discuss this topic, but Yusuke Jinnai seems to be making progress on the question. Here is part of his abstract:
In this paper, I propose a new empirical approach to identify the impact of charter schools on local traditional schools. Specifically, I define direct impact as the effect of introducing charter schools on traditional-school students in grades that overlap with charter schools’ grades, while indirect impact is defined as the effect on students in non-overlapping grades. Unlike prior research work, which estimates the effects of charter school entry at the school level, I examine the impact at the grade level by exploiting the variation in gaps between grades offered by charter schools and grades at nearby traditional schools.
Using student-level panel data from North Carolina, this paper shows that the introduction of charter schools does not induce any significant indirect impact but generates a positive and significant direct impact on student achievement. Distinguishing between the two distinct impacts and taking into consideration both traditional-school and charter-school students, my study finds overall positive effects of introducing charter schools on student achievement. I also demonstrate that such overall effects would have been underestimated by 85% in the literature, since previous work identifies the impact of charter school entry at a moment when the direct and indirect impacts are likely to be mixed.
Finally, I argue that the direct impact consists of student sorting effects and competitive effects and, by controlling for unobserved peer characteristics, demonstrate one-quarter of the positive direct impact is driven by the former while three-quarters result from the latter.
The paper is here, and Yusuke is on the job market from Rochester this year. His entire portfolio of papers on education appears to be quite interesting.
Here is a related post on school choice in Sweden, from Modeled Behavior.