In Arthur C. Clarke's story "Into the Comet" he describes a spaceship with a computer malfunction that dooms all abroad to eventual death by starvation/oxygen deprivation, whichever comes first. The solution is a device older than the computer: the abacus. The entire crew ran calculations on acabi, and they make their way out of the comet's nucleus successfully. That is an extreme example of citizen science (or oh-my-God-we're-all-going-to-die science) but it shows the principle, that collaboration by a large number of people can solve very complicated problems. Michael Nielsen's excellent book, 'Reinventing Discovery' tells us about many such examples, though in most of them participants have to do a lot more than just calculate without thinking.
Another story Nielsen recounts is the story of the Polymath Project: Fields Medal recipient Tim Gowers posted a mathematical problem in his blog and asked for a collaborative efforts. Twenty-seven people wrote 800 comments and solved the problem within 37 days. Now there is a Polymath blog which keeps up the good work.
These projects were a success, but Nielsen also studies failed projects and the reasons for their failure. He argues (which I wholly agree!) that scientists are rewarded by writing as many good scientific papers as possible. Contributing to, say, Wikipedia, essentially takes away time from research and gives nothing in terms of academic reputation.
Galaxy Zoo is a success because it gives astronomers something to write about, and it's possible the Polymath project succeeds because it A. involves people with tenure and B. involves people who want to be noticed by people with tenure.
Personally, I think the solution to scientists' reluctance to cooperate in collaborative projects is simple: put them in a spaceship and tell them they won't be able to make it home until they collaborate. However, it is possible the oxygen run out while they'd argue about whose name gets to be first in the authors' list. Also, spaceships are very costly.
Next part: what Nielsen has to say about Arxiv and the future of open science.
Michael Nielsen talks Open Science in a TED event:
Nielsen, Michael (2011). Reinventing Discovery Princeton University Press Other: 9780691148908