Install SSH servers and clients on both server hosts. Then use a CPAN module like Net::SCP or others to securely copy files between servers.

I have never used one of these modules myself yet, though they look very well documented and easy to use.

I have used rsync, which Arunbear suggested, in the past and it worked well.

Now I usually set up password-less SSH on each server and then use git or scp from the command-line.

To setup password-less SSH (after you have installed an SSH client and server on both hosts):

  $ cd ~/.ssh
  $ ssh-keygen -t rsa
[download]

You will be prompted to enter a filename in which to save the key (~/.ssh/id_rsa): <thishost>on<targethost>

  $ ssh-copy-id -i <thishost>on<targethost>.pub <targethost>
[download]

or for Macs or other OSs which do not have ssh-copy-id:

 <targethost>$ cd ~/.ssh
 <targethost>$ ssh <thishost> cat ~/.ssh/<thishost>on<targethost>.pub 
+>> authorized_keys
[download]

Test SSH with Identity Key:

  $ ssh -i ~/.ssh/<thishost>on<targethost> <targethost>
[download]

Lastly, create file ~/.ssh/config

  Host <targethost>
  Hostname <targethost>
  StrictHostKeyChecking no
  User <Your username>
  IdentityFile ~/.ssh/<thishost>on<targethost>
[download]

The beauty of the config file is, I can ssh into any one of my hosts or mobile devices, regardless of OS, without providing username or password. CAUTION: You MUST carefully protect access to any host on which you use password-less SSH.

UPDATE:

Several years ago I used Sparkleshare to sync between desktop hosts/servers, in conjunction with rsync on iOS devices (jailbroken), to keep personal files synchronized. It worked pretty much OK back then. You might want to check it out if your files in question are larger than what Dropbox or iCloud. etc. can handle or if git is not suitable.