The "Putty" tools (in Windows) are basically Windows-compatible implementations of a set of standard Linux/Unix tools: [font=courier]ssh, sftp,[/i] and so on.
As far as I know, you do have to be "on the same network," but this is simply in the sense that there must be some direct path between your computer and the computer that you are trying to connect to. But, "that's true of TCP/IP networking in general."
As far as the rest of your question ... why, "that is what the Internet is for." You are indeed asking (whether you yet realize it or not...) a vast question. Indeed, several questions, all of them well beyond the scope of what we could possibly tackle here.
The first part of your question has to do with [i]network file system support," in the sense of your Linux box being able to recognize that a particular network file system (and there are a great many...) is being used.
The second part, though, has to do with single sign-on, where many different computers in a network understand and accept the same set of authentication credentials and from them provide compatible and consistent authorization. Microsoft uses something called "LDAP" for this, having brazenly re-named it "Active Directory" as they are so prone to do.
It's a vast subject... too big to tackle further, and I won't try. But I've just given you a lot of great "Google keywords."