I usually use WS_FTP (Windows) to transfer files from one server to another (without having to first transfer them to my machine) but I don't have it here and would prefer to use a free tool. Doesn't look like it's possible in FileZilla. Any other suggestions?
What you are referring to is commonly known as "FXP" (File eXchange Protocol [sic], even though it's not a protocol, just a feature of FTP), and I was under the impression that most modern FTP servers disallowed it as a security feature, thus rendering support for it in FTP clients somewhat lacking as of late.
A free solution is BitKinex, which is now freeware and development has stopped since march 22, 2009:
BitKinex integrates the fuctionality of an innovative FTP, SFTP and WebDAV client for Windows. In addition to features found in other popular FTP programs (like support for the SSL/SSH, multipart and multithreaded transfers, remote edit or FXP) our FTP client introduces several unique approaches and solutions.
Note: I've never used this FTP client and cannot vouch for it, but it's the only free solution I found. Commercial products include SmartFTP, FTPRush, CrossFTP, FlashFXP.
SmartFTP supports FXP, but it's not free.
You can download a fully-featured trial version from their site.
I'm quite sure there is no desktop client solution for this as it is not possible to do with the FTP protocol. I have used a PHP script and PHP's FTP functions to recursively and directly copy from one server to another where I had no shell access, however the successful completion of such an operation depends on many server settings like the maximum execution time for PHP.