I think it needs to be done as soon as possible. It's an important skill, and letting it go is asking for trouble.
The issue of how much contact is best answered, IMO, by "What is the student capable of?" It has to be about their comfort level at the beginning levels. I'm one of those people who is now having to rewire my fighting style because of prior bad instruction. You can't just dump a new student into the ring and say "spar", and wonder why they get upset after getting beaten down, and sadly that is how some schools do things. They need to develop thier comfort zone.
However, you have to be careful of the other extreme as well. Students need to understand that we do martial arts, and that part of what we do entails physical contact, sometimes of the rough kind. We're not pillowfighting, and we're not doing ballet. You are going to get hit, and students need to be able to understand that. There is nothing worse than someone who has a rank or two, has sparred, and gripes that "you hit them too hard". My ex does TSD and has brown belts and red belts complaining that they actually have to work when they deal with him. And his control has been noted many times by the instructors, people just get lazy. I'm not advocating brawling, because not every student wants that, but if you have achieved a certain level and still aren't comfortable getting hit, I have some question as to your skill and why you were advanced in the first place.
So, we do lots of live drilling in my school. One of the first techniques people are taught involves a throw. We ask that they actually perform the throw to get them used to actually engaging their opponent. We do a number of different drills with pads, movement, pushing hands, and others to get people used to throwing punches and kicks at each other, as well as defending against them. We also have what we call "touch tag". The goal is to tag your opponent on the knee or shoulder. It works a lot of the same skills as sparring, and is a good way to keep students on their toes in-between sparring sessions.
I would close by stressing that the last sentence above is very important. Drilling is necessary to work the skills needed for sparring if your school doesn't spar regularly. This way students don't get panic attacks when the sparring gear comes out because they've only ever hit a stationary bag.