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Thank you for your comment. We agree that education is a fundamental human right and that each local, regional and national government should ensure learning for all. Each child has a right to a good quality education regardless of whether the provider is public or private. The political context is important and something that has been mentioned by others above which we reaffirm here.    Rigorous impact evaluations are mentioned in the article with links but far more is needed as you point out in terms of rigorous research in the area particularly in different development contexts. The Bank is working to do this through its Strategic Impact Evaluation Fund (SIEF) and there is a growing academic literature also. Its a space which offers innovations and opportunities which need to be balanced by strong government stewardship. On the evidence...

Charter type schools 

Abdulkadiroğlu, A., Angrist, J. D., Dynarski, S. M., Kane, T. J., & Pathak, P. A. 2011. “Accountability and Flexibility in Public Schools: Evidence from Boston's Charters and Pilots.” The Quarterly Journal of Economics, 126(2): 699-748. 
Showed that charter schools who were subject to greater accountability had large and significant test score gains for lottery winners in middle and high school. In contrast,  for pilot schools, who were subject top less accountability than charters,  gains were small and mostly insignificant for pilot school lottery winners in terms of  test scores. 

Angrist, J D. Dynarski, S.M., Kane, T.J., Pathak, P.A. and Walters, C.  2010.  "Inputs and Impacts in Charter Schools: KIPP Lynn." American Economic Review Papers and Proceedings 100(2)
Found that charter school winners performed 0.4 s.d higher on math than charter school losers and that KIPP Lynn raises achievement more for weaker students.

Dobbie, W., Fryer, R. G., & Fryer Jr, G. 2011. “Are High-quality Schools Enough to Increase Achievement Among the Poor? Evidence from the Harlem Children's Zone.” American Economic Journal: Applied Economics 3(3): 158-187
Found the effects of attending an Harlem Children's Zone middle school are enough to close the black-white achievement gap in mathematics. The effects in elementary school are large enough to close the racial achievement gap in both mathematics and English Language Arts. 

Hoxby,C.M. and Murarka, S.  2009. "Charter Schools in New York City: Who Enrolls and How They Affect Student Achievement." NBER Working Paper 14852.
Found that charter schools performed 0.09 standard deviations per year in math and 0.04 standard deviations per year in reading with a longer school year related to higher student achievement .

Tuttle, C. C., Gill, B., Gleason, P., Knechtel, V., Nichols-Barrer, I., & Resch, A. 2013. “KIPP Middle Schools: Impacts on Achievement and other Outcomes.” Washington, DC: Mathematica Policy Research.
An evaluation of 43 KIPP middle schools found an average estimated impact of 0.36 standard deviations in math. 

Evidence from developing countries 

Barrera-Osorio, F. and Raju, D. 2011. "Evaluating Public Per-student Subsidies to Low-cost Private Schools: Regression-discontinuity Evidence from Pakistan.” World Bank Policy Research Working Paper No. 5638.
Found the program has also increased test scores in math and science by 0.4 and 0.5 standard deviations.
French, R., and Kingdon, G. 2010. “The Relative Effectiveness of Private Government Schools in Rural India: Evidence from ASER Data.” Department of Quantitative Social Science Working Paper No. 10-03. Institute of Education, University of London. 
Showed that the private school achievement advantage of 0.17 standard deviations.

Muralidharan, K. and Sundararaman, V. The Aggregate Effect of School Choice: Evidence from a Two-stage experiment from India. NBER Working Paper No. 19441

The results suggest that private schools in this setting deliver (slightly) better test score gains than their public counterparts, and do so at substantially lower costs per student. 

The Impact Evaluation in Education database includes many more studies on the private sector and can be accessed here