In order to be able to dowload/upload from/to any computer, of course, you need to be in the same network somehow (if not directly, then a router or bridge should connect you to a second network)
So if you want people from internet be able to dowload the files from your local server, you need to connect it somehow: either create a remote connection to the server where users can log in using username and password, or a VPN, or connect the server to the internet and install a FTP server software.
You may also use services like dropbox or cubby. These services are, precisely, to be used to host and share files. You have a free option as well as a paid one, and that may serve to your purposes.