You don't need to download a local copy, if you use the Content Server that is part of calibre. But you do need to set up port forwarding and ideally a Dynamic DNS service like
https://www.duckdns.org/ if you want to access the server outside your home WiFi network.
The other option is to use calibre2opds (see my signature), usually with Dropbox. Although it may be possible with Copy.com, IF they allow you to share a folder and make it into a pseudo-server.
I've got lots of Dropbox space, mostly because people signed up via my referral link in my signature.