You should seriously consider a source control system, which is the best solution here. That gives you one master copy of the code on the network. Each computer retrieves the latest revision from source control, making a copy on the local disk. Source control helps maintain a known good backup, and allows you to track changes and compare versions or revert to older code when something breaks.
I'm fairly certain you can open the same VIs stored on a network drive on multiple computers concurrently. Global variables are irrelevant here. Each computer will independently load the VIs into memory. The problem is if you edit and save the VI on one computer while it's open on another, and if you make edits on both simultaneously, you'll definitely lose some of the edits and could possibly corrupt your VI and lose all of them, which is why you should be using a source control system instead.
If you have the application builder, I recommend building your code into an application. You can store the application the network if you want, and run it on multiple computers.