When vouchers are used for private schools, the public doesn't have much of a say as to how the money is spent. That's probably not a big deal, most of the time, since most private schools run in a similar manner to public schools, minus the red tape. One problem that does exist, even among people aligned with the private school, is oversight. Those who make the rules are often making rules that benefit themselves, and most people involved don't even know how to be part of the system in order to ensure there isn't too much self-serving going on. 

One example I have is a private, religious school that has a school board made up of church members, while the school population is 90% non-church members and 90% on vouchers or scholarship. So we have parents using state money to send their kids to a school that neither the parents nor the state have much say about the decisions being made. In fact, most church members and teachers don't know who are on the school board, while some school staff have relatives participating as members.  This isn't a big deal until something negative happens, like a teacher being accused of not following school board policies: if that teacher has a spouse on the board, no big deal; if not, big deal. Or maybe a board member / church member would like a relative employed at the school. 

The main point is that parents should understand the structure of the school they plan to choose. Maybe some parents prefer a school board made up of church members rather than fellow parents or community members who won an election. In general, school boards are made up of non-teachers who don't really understand what it takes to be a teacher, and that goes for private schools, public schools, and charter schools.