I don't know much about the Downs, but I do know that where I live, coastal Suffolk, there are dozens of places where it's possible to wild camp without being challenged or even noticed. This area is very much like the Downs, just without any hills. My favourite spots are concealed corners in birch thickets or on the heaths.
In fact, I once came across a bright red tent pitched within ten yards of a fairly well-used path in Tunstall Forest. Wooded areas are generally best because there's a lower likelihood of being spotted. I have two or three really excellent bivvying spots within ten minutes walk of my house--unsuitable for tents, but there's something very satisfying in building your own shelter and cooking your food over an open fire.
In the three years since I constructed the first bivvy site, I'm fairly certain nobody else has ever come across it. It's invisible from the path and can only be accessed by scrambling through thick bracken and birch scrub.
So my advice would be to stick to out-of-the-way, wooded bits. And as someone mentioned above, all water has to be carried with you, which can make long camps difficult.
