There is a new NBER working paper by Richard J. Murnane, Marcus R. Waldman, John B. Willett, Maria Soledad Bos, and Emiliana Vegas. I have not had a chance to read it, but here is the key part of the abstract:
We found that:
1. On average, student test scores increased markedly and income-based gaps in those scores declined by one-third in the five years after the passage of SEP.
2. The combination of increased support of schools and accountability was the critical mechanism through which the implementation of SEP increased student scores, especially in schools serving high concentrations of low-income students. Migration of low-income students from public schools to private voucher schools played a small role.
We interpret these findings as more supportive of improved student performance than other recent research on the Chilean policy reform.
That is not exactly the Milton Friedman story, but it is essentially a positive report for vouchers.