I love your point about the difference between when people willingly give to an individual versus when government takes it from everyone and distributes it how they see fit. The biggest difference is the feeling of compassion on the one side and gratitude on the other. That's such an essential part of what makes giving worthwhile.
This idea definitely has merit and I could see it as better than the current system because of the fact that there would be no centralized power governing who got money or not.
One problem that I see though, is that since you wouldn't personally know the people requesting money, it would be hard to tell when people were making up stories about what they needed the money for. It might become more of a "who can dream up the most elaborate story" contest rather than the funds going to the people who actually needed it the most who maybe weren't as eloquent in their presentation of their request.
The only thing I can think of to solve this would be to have some kind of verification process to validate the user and their request. But I don't know how you would do that without a centralized judging committee. I guess there could be some kind of a pre-voting process done by those who had proven themselves to be contributing members of the community. But I'm not sure how that would work.
What starts out as a simple and brilliant idea can quickly become a complicated mess just by virtue of the fact that we can't fully trust one another to be honest. In a perfectly trustworthy society, I think your idea could be a perfect solution.