Can (and Should) Public School Teachers Teach Private School Students (In RI)?

The biggest issue regarding The Grace School Academy application should be the question of whether or not public school teachers can or should teach in classrooms of mixed public and private school students. To quote the application:

At full maturation, each classroom will have approximately 20 students; 17 from The Grace School Academy who will enroll through the lottery. These students will be joined by approximately three (3) Meeting Street School students.

One would hope that this is just illegal, but I'm not going to bother figuring that out. It appears that in the budget figures for the school that the cost of classroom teachers is shared between the public and private school budgets, but I'm not really sure how it works, and it isn't clearly explained at all, which is pretty weird considering I've never heard of anyone trying such a thing anywhere before. Seems worthy of some elaboration.

And even if in this case it is legal and actually makes sense, I would still want to see some very clear standards for what is and is not permissible for this kind of situation, because if lots of private schools decide they want their own embedded/integrated charters for some reason, things could get out of hand quickly. This could turn into a kind of de facto voucher program.