To copy files between the two FTP-servers (accounts) directly without downloading the files to the local PC, you need to turn on the pure-ftpd option of the FTP-server — AllowUserFXP.
In the configuration file of the FTP-server /etc/pure-ftpd.conf, you need to write ‘yes’ in the AllowUserFXP parameter. Then you need to reload the FTP-server:
AllowUserFXP yes
service pure-ftpd restart