Also, how much space should I leave between the fence and the support structure? How much space does the plant need?

IME, SAC needs no horizontal supports. It will easily grow to the top of most fences (5-6' height or more) with little to no assistance. And then quite rapidly extend along the top of the fence in both directions. And because it is such a big and vigorous vine, it drapes down as well, effectively covering the fence quite handily. If you did want to train it more laterally along the face of the fence rather than just letting the fence itself support it, I would space any horizontal supports at least 12" apart, perhaps more.

This is a big, rampant vine that can swallow a small car!! It can extend 25-30 feet when established.

Thanks for the info. I do intend to train it laterally along the fence, because I'm looking to hide both the view of our neighbour's house and backyard. Also, since the fence is just large wooden boards, without some sort of support to grab on to it will just tangle along the ground.