I've seen schools use a variety of approaches. Some, like the Kyokushin schools, let their white belts do full contact free sparring. Others might not even let their people free spar until they hit brown belt of higher. One system may say "it's better for them to experience free sparring early, so their techniques can get better from it," while another many say "it's better to wait until their techniques are refined enough."
You know what? It works both ways, provided that the teacher knows what he's doing, and provided that the students listen. Now, some students are going to do better in one situation than another. To each his own.
I prefer letting people wait until they've at least started their intermediate classes (10th-8th kyu white and yellow belts are in the beginners class, 7th-4th kyu orange through blue belts are intermediate students, 3rd kyu brown and up are advanced). At this point, their fundamental techniques are usually refined sufficiently.
Even then, their first few sparring sessions are going to be against senior brown belts, or even the black belts, who are there to act as mobile targets that put up a simple defense. As the student grows more experienced, the mobile targets then start offering some simple attacks. Eventually, we start pairing the students up against similarly ranked students, under the direct watchful eye of the instructors.
I prefer this method, since it gradually builds up confidence in the students, and there's fewer injuries along the way.