> Has anybody put together a project to build a "distributed observatory" to coordinate the efforts of a great number of amateur astronomers, each with a small telescope?

For the purposes of looking for faint objects, not that I'm aware of. But there is an effort called the "Center for Backyard Astrophysics"[0] that is a loose network of (mostly amateur) observatories to do long time series brightness measurements of variable stars. I was involved in taking some data with/for them ~10 years ago while an undergrad[e.g., 1] and it was a fun experience.

[0] http://cbastro.org/

[1] http://spiff.rit.edu/richmond/ritobs/sep18_2005/sep18_2005.h...

Yes, for example: Project PANOPTES[0] which aims to use many citizen scientists' participation to discover transiting planets. Check it out!

[0] http://panoptes.github.io/