It depends on what system and what application you're doing the move from.
If you're doing it via the machine where these shares are at, it'll transfer locally through the OS. Example: ssh to server and using the MV command or opening a File Browser and transferring files from one File Browser to the next (so long as both File Browsers are opened to resources on same server).
The reason for this, if you're using two File Browsers on a remote session, is because whatever you do on your end, gets translated on the server end first. The server looks at the request and determines the shortest path to the destination and since it see's itself as the source and destination, the traffic never leaves the server (outside the updated tracking info you see for the file transfer).
If you're doing it through a complex remote session through a few computers, where you forcefully tunnel the traffic, then the transfer will happen over the network connection.