Can I transfer files between two remote servers with WinSCP?

To transfer files between the two servers via local workstation
To transfer files directly from one server to another

To transfer files between the two servers via local workstation

See documentation to learn how to duplicate files between sessions.

To transfer files directly from one server to another

This feature is not built into WinSCP, for reasons listed further down. Yet, there are ways to utilize some advanced features of WinSCP to achieve the task.

Pushing files with custom command

With SFTP and SCP protocols, you can push the selected remote files to another server using scp custom command.

Pulling files with remote command

With SFTP and SCP protocols, you can use suitable remote command to download files to the remote server from another server. E.g.:

wget ftp://username:password@otherhost:/path/file ./

Reasons why the feature is not built into

SFTP protocol does not allow this.
With SCP protocol it may be theoretically possible. However it has many drawbacks. It would require not only major changes in WinSCP, but also many configuration changes on both servers to allow authenticate on the second server from the first one (note that it would not be possible to use an interactive password authentication). See scp custom command for details. So practically it would have advantage for advanced users only, who can use the “custom command” solution easily.
FTP protocol has an FXP extension that allows this, but it is not supported by WinSCP. 309