Yes, it's pretty common. The downside to these kinds of arrangements is that the family potentially is not able to do a lot for the kid. If the doctor or the school system decides to ask for any paperwork you're not technically able to take the kid to the doctor, make educational choices, etc.
true, but whether that is an issue really depends on how involved and cooperative the parents are. presumably if parents are looking for private foster places they are able to be involved when needed, like register kids for school. the doctor should also be manageable, otherwise i could not even send my kids to their grandparents for the holidays. so i think in most cases this should work out. no doubt however there are cases where it doesn't. but in such cases government involvement is probably warranted anyways.