There are several ways to do this in Linux. The two ways I find easiest are as follows:
1. NFS. NFS is the native *nix network file system which allows you to mount remote directories as if they were local parts of your filesystem (think mapped drives in Windows.) Depending on how you want to set it up, you can run NFS servers on each Linux machine or just one to swap files back and forth.
Here's an online howto: http://nfs.sourceforge.net/nfs-howto/
2. SSH. SSH is the secure transfer protocol which replaced telnet as the preferred way to access remote systems, since everything in SSH is encrypted. If you're running sshd on each of your Linux machines, it's as simple as using scp (secure copy) to copy files back and forth. It works like this:
scp filename hostname:/directory/directory
For instance, if I was copying a file called example.txt to my second computer called moloch, it would be:
scp example.txt moloch:/home/chris
scp is great for doing individual copy operations, especially when you're copying things like .html files and what not to a remote web server. NFS is easier for when you just want to share files within your own LAN.
Hope that helps!