We would need an absurdly diverse and large group of people for this to be effective on a large scale. If it was just the hacker/designer community working on collecting data then we could potentially have a great amount of data for those specific niches and perhaps a bit of data for random other niches.
Building the plugin itself would not be too challenging, but it would require "fixing" every time Twitter changes any little thing in their layout.
I think I much prefer the idea of having a massive botnet, each node with its own API keys, querying their own subsection of Twitter to build one database. In the spirit of collaboration, it could be possible to write a program that would carry out all of these tasks in a distributed manner on each person's individual computer. If you have one source of accounts + api keys + proxies that were auto distributed to each node you could keep each machine working.