The technology they're using to make this happen is called NFS - Network File System. They may additionally be using another technology with NFS called automounts, specifically Autofs if it's Linux.
NFS
NFS allows folders from one machine to be accessible to another. There is nothing magical about this. There is a client and server that is facilitating this connection. When you access one of these directories that's been mounted on your system via the NFS client, calls are made from your system to relay the information regarding the directory and it's contents.
If one of the clients accesses a file within the directory, the NFS server relays the contents of this file to the NFS client too.
Autofs
Automounting is a technology that allows a client system to access a shared remote resource, such as NFS, on a temporary basis. That is to say that the NFS client system has access to this remote NFS share. But it isn't actively "using" it until someone attempts to access the shared directory. Only then does the NFS client attempt to "mount" this remote directory and its contents.
File content life-cycle
In either case there is no implicit transfer of the files to the NFS clients. That is to say they aren't physically copied to the clients in any long term way. The files are streamed to the clients when they attempt to access them. After use they're gone; they do not persist at the clients in any long term form.
Just to give you a rough idea, you could use the program strace to see some of the system calls that are made as a program runs. So, using this command for example:
$ strace echo "hello world" > afile
We could see how a file is written to the system. We could also use this form to write to an NFS mounted directory:
$ strace echo "hello world" > /home/sam/afile
Both of these traces a virtually identical.
### Local write
$ cat strace_wr_local.log
execve("/bin/echo", ["echo", "hello world"], [/* 33 vars */]) = 0
...
fstat64(1, {st_mode=S_IFREG|0644, st_size=0, ...}) = 0
mmap2(NULL, 4096, PROT_READ|PROT_WRITE, MAP_PRIVATE|MAP_ANONYMOUS, -1, 0) = 0xb7f80000
write(1, "hello world\n", 12) = 12
close(1) = 0
### NFS write
$ cat strace_wr_remote.log
execve("/bin/echo", ["echo", "hello world"], [/* 33 vars */]) = 0
...
fstat64(1, {st_mode=S_IFREG|0644, st_size=0, ...}) = 0
mmap2(NULL, 4096, PROT_READ|PROT_WRITE, MAP_PRIVATE|MAP_ANONYMOUS, -1, 0) = 0xb7fbe000
write(1, "hello world\n", 12) = 12
close(1) = 0
Reading is only slightly more interesting, but still basically identical.
### Local read
$ cat strace_rd_local.log
execve("/bin/cat", ["cat", "afile"], [/* 33 vars */]) = 0
...
fstat64(1, {st_mode=S_IFREG|0644, st_size=1761, ...}) = 0
open("afile", O_RDONLY|O_LARGEFILE) = 3
fstat64(3, {st_mode=S_IFREG|0644, st_size=12, ...}) = 0
read(3, "hello world\n", 4096) = 12
write(1, "hello world\n", 12hello world
) = 12
read(3, "", 4096) = 0
close(3) = 0
### NFS read
$ cat strace_rd_remote.log
execve("/bin/cat", ["cat", "/home/sam/afile"], [/* 33 vars */]) = 0
...
fstat64(1, {st_mode=S_IFREG|0644, st_size=1771, ...}) = 0
open("/home/sam/afile", O_RDONLY|O_LARGEFILE) = 3
fstat64(3, {st_mode=S_IFREG|0644, st_size=12, ...}) = 0
read(3, "hello world\n", 4096) = 12
write(1, "hello world\n", 12hello world
) = 12
read(3, "", 4096) = 0
close(3) = 0
Wading into the pool
If you're just generally curious about the NFS protocol you can read more about how it works here, in the section titled: NFS protocol basics. It's generally easy to get the basic concepts of how it works and there is an example of an NFS request, just to give you a general idea of how things work.
Diving deeper
If you truly want to peek behind the curtain you'll likely need to bring in a set of tools for collecting network traffic so that you can see the flowing of bits back and forth between the NFS server and one of its clients. Tools such has tcpdump or wireshark will likely be your friends in doing this deeper dive.
I would caution you to not waste your time unless you're truly the curious type, since deep dives like this require much skill and familiarity with a suite of Unix tools that I would consider only someone who has been using Unix for a dozen or so years.
This site will help you with this endeavor if you're truly curious:
