Ideally it would be a Github repo that anyone could use to if they wanted to add a feature, or tweak current features.
Could this actually work in theory? How would you stop malicious intent? How would you deal with securing data from potential changes?
On the infrastructure side, there have been attempts at self-hosted wikis, where the files that run the site are editable from within the site itself.
Unfortunately, the "could this actually work" part isn't a technological problem, it's a sociological problem. You can't just put something out there and expect it to magically go if you define enough software rules. The hard part of any community isn't the software, it's the community management.
LambdaMOO tried letting its users self-govern and they just ran amok, and admins had to come back in and take control.
Today, you see community-oriented startups like Cowbird have community managers as cofounders or first employees. Flickr's cofounder Caterina Fake famously posted well-written, polite comments on every single photo posted to Flickr for hours every day, for the first N months or years, in order to set the tone for all the users that would follow her.
There's a new textbook out, Building Successful Online Communities, which seems to have done a great job in surveying the state of the art in online community building: http://successfulonlinecommunities.com/
Because it's really a sociological problem, the correct answer to this is to dig into fifty years of sociological research, starting with Bion's "Experiences in Groups." Clay Shirky pokes at it pretty well in his piece, "A Group Is Its Own Worst Enemy": http://www.shirky.com/writings/group_enemy.html
Groups of humans need hierarchy. There are thought experiments about how you could make a group function otherwise, like Block's "Community: The Structure of Belonging", but they're unproven.
For a long list of references, you can read my references for an essay on community organizations, which starts on page 2 of this PDF: http://distance.cc/issues/01/vitorio-colophon.pdf
i'm now looking to do this on a larger and more public scale --- i'd love to chat with you more about what you are trying to do and we could talk about possible best practices - perhaps this is the next evolution of the internet?