rsync only "syncs up" files from one server to another. The difference is that it does it more efficiently that copying all files, and all parts of large files - just those parts of data which are different between the two servers are transmitted.
It sounds like what you need to some sort of distributed merge system. I would go about it by choosing "master" server to pull files from all the other servers, write a custom program to merge all these together into a master list, and then push them back out to the other servers.
The fetching of the partial files and pushing out of the new merged file could be done using rsync, although it has quite a large memory overhead when there are a lot of files, so if you don't have a massive amount of data, you might be better off just using scp or something like that.