The commonest way to achieve this as far as downloading ebooks is concerned is to use the Calibre Content server to give online access to the Calibre library.
If you do not want Calibre running on the machine that is doing the sharing then it might be worth looking at
calibre2opds (possibly in conjunction with Dropbox) as an alternative approach.
As far as uploading books is concerned, then Dropbox is a good way of letting many people have a shared folder to which files can be added to later be added to the calibre library.