This sounds like you'd be interested in a special FTP usage called FXP.

However, since this is prone to vulnerabilities, it is highly probable that the admins of server2 and server3 disabled it (or, depending on server system, didn't explicitly enable it).
If they are fine with it (e.g. because it's an internal network), it should be possible to do it with $ftp->pasv and $ftp->port commands.