Is it possible to FTP files from a remote server to another remote server, or are you required to first download the files to local folder, then upload from this local directory?

FTPeek gives you a set of functions like a standard FTP client. It would be possible to have a script run one one server machine and have it send files to another server machine, but as with all FTP communication one machine is acting as a "client" and one as a "server." Files will always need to be located on the machine that they are coming from, even if they are transferred to it just temporarily before being uploaded to the destination machine.

If you're using remote desktop software to log into a machine and then run an FTP script from the remote machine you wouldn't need to transfer the files to your local machine.

You would not be able to transfer files from FTP Server A to FTP server B with Client C acting as the FTP client without downloading from A and then uploading to B. It would be possible to have server A connect to B directly and then transfer files, but you couldn't use C to transfer from A to B without the file going to C first.