I think a lot of the private placement of out of state kids is kids in foster care who have special needs.
There are over 1000 kids in out of home placements.
https://cfsa.dc.gov/sites/default/files/dc/sites/cfsa/page_content/attachments/Children%20and%20Youth%20CFSA%20Statistics%20July%202016_0.pdf Half of all DC foster placements, and probably more of the older kids, are in Maryland. It wouldn't surprise me if most of those kids had such severe needs that they required private placement. That could easily be 300+ kids.
Then there are the kids who are supervised not by CFSA but by DYRS.
I'd be glad for someone to look into it but am not sure it's the next scandal.
I'd go with: bad accounting by PTAs, corruption in construction contracts (like dragging out timelines and then charging extra for expedited work), or discrepancies in accepting mid-year placements or OOB kids (why do some DCPS schools take kids in upper grades if there's space and others don't?).