Another possibility is to set up a shared directory with the sticky bit set.
On a Linux system, the /tmp directory has the permissions drwxrwxrwt, or 1777 in numeric terms, which ensures that anybody can write anything in there, but once that do those files belong to them and can't be modified or deleted by other users, so you maintain a concept of ownership of the files.
So, you can create a directory with these same permissions as a kind of group directory.
This doesn't exactly work in the way you describe above, but it does ensure that users can put whatever files they like in there, into a new directory which they themselves create or just into the root, and no other user can delete or modify any files belonging to other users. The sticky bit on the directory is what achieves the second part - normally if you give a directory world-writable permissions, then people can delete the files of others. With the sticky bit, people cannot delete or modify files not belonging to them.
For any common files that you want anyone to be able to read, you just put it in there yourself and leave it owned by you.
Note: unlike in your question, other people won't be able to modify or delete these files, but they would just need to be instructed that they can set up their own copy wherever they like and modify that instead.
The downside to this is that they can't transparently delete or modify whatever they like if it doesn't belong to them.
But the upside to this solution is that everything they put in there is shared with everyone else, who can all open it and read it.