That's not a Java question, so I'm going to move it to a more appropriate forum.
There's no one answer to it, however, since there are many remote desktop products and they have different abilities and options. Some can copy as simply as doing a "drag and drop" from the local desktop to the remote one. Some cannot do remote file copying at all.
For automated production environments, you wouldn't even be using remote desktops. There are remote file-copy utilities like the scp and rsync programs. They operate from command line or scripts and don't have GUI interfaces.
Loudly announcing something is true and finding out you're wrong makes you feel foolish.
Finding out you're wrong and refusing to admit it makes you LOOK foolish.
Well, it depends on what type of remote connection and application is in use.
As you say "from remote A to remote B" I guess there's also a local client C involved which connects to both remotes separately. For the most simple solution an interesting question is: Is a direct connection from remote to the other possible? If so, no matter which direction, you could use well established protocols like SSH and SCP, which builds ontop of it (it's like a plain remote copy but secured by the SSH tunnel) - or even set up a webserver and use regular HTTPS. Another option is setup a VPN and see both systems as if they were within the same local net.
If both remotes can't directly connect to each other - maybe you can set up somthing like STUN - use a rendezvous server which can be reached from all machines. This also depends on firewalls in place if they allow hole punch.
As Tim already said: there's no "the one and only" answer to it - it heavily depends on what is available/possible.