This is one of those scenarios where the concept of "rights" makes me nervous. As in, "Children have rights, don't they? And shouldn't there be some entity that ensures they have them?"
I'm pretty doubtful there are such things as rights. If not, then the problem is solved.
Well, what about child abusers? How are they handled in an anarchist world?
I say, handle the problem yourself. That is the problem with "hiring" the state to fix things: every little thing will become a major event. Spank your child? Get a visit from CPS, and so forth. And the person who reported you endures no consequences.
In an anarchist world (just my opinion), it falls back on the person who would have been the accuser. That person must either get personally involved, or mind his own business and leave it alone. These facts obviously mean there will be far fewer interventions for trivial "abuses", while there is some path still to prevent larger abuses (e.g., "stop beating that kid or I'll kick your ass the same way").
Will some kids "fall through the cracks"? Sure. That's a given. Human beings are not capable of perfection. It is only a socialist fantasy that people can be perfected by a caring state. If you think the state is perfect, think of those 400 Mormon kids in Texas.
Parents own their kids. Period. If you don't think so, try taking mine from me, see where it gets you.