[This post is written for the non-mathematician. So if there's something that needs clarification, please ask!]
It seems to me that, at least in theory, a different model could work: different, that is, from the usual model of people working in isolation or collaborating with one or two others. Suppose one had a forum (in the non-technical sense, but quite possibly in the technical sense as well) for the online discussion of a particular problem. The idea would be that anybody who had anything whatsoever to say about the problem could chip in. And the ethos of the forum — in whatever form it took — would be that comments would mostly be kept short.
Dr. Gowers chose a problem in mathematics that he thought amenable to an online collaborative approach, then kicked things off with a blog post. Six weeks later, the main problem he proposed was declared (essentially) solved. However, the project still continues apace, especially at threads at Terry Tao’s blog.
I’m going to explain the main idea of what Polymath accomplished in as comprehensible a manner as possible.