So I have tried to find a answer but must not be searching correctly or what I'm trying to do is the wrong way to go about it. 
So I have a simple python script that creates a chess board and pieces in a command line environment. You can in put commands to move the pieces. So one of my co workers thought it would be cool to play each other over the network. I agreed and tried by creating a text file to read and write to on the network share. Then we would both run the script that reads that file. The problem I ran into is I pretty much DOS attacked that file share since it kept trying to check that file on network share for a update. 
I am still new to python and haven't ever wrote code that travels the internet, our even a simple local  network. So my question is how should I go about properly allowing 2 people to access this data at the same time with out stealing all the network resources?
Oh also im using version 2.6 because thats what everyone else uses and they refuse to change to new syntax
First off, without knowing how many times you are checking the fle with the moves, it is difficult to know why the file-share is getting DoS-ed. Most networks and network shares these days can handle that level of traffic - they are all gigabit Ethernet, so unless you are transferring large chunks of data each time, you should be ok. If you are transferring the whole file each time, then I'd suggest that you look at optimizing that.
That said, coming to your second question on how this is handled at a network level, to be honest, you are already doing it in a certain way - you are accessing a file on a network share and modifying it. The only optimization required is to be able to do it efficiently. Even over the network operations in a concurrent world do the same. In that case, it will be using fast in-memory database storing various changes / using a high-scale RDBMS / in the case of fast-serving web-servers better async I/O.
In the current case, since there are two users playing the game, I suggest that you work on a way to transmit only the difference in the moves each time over the network. So, instead of modifying the file over the network share, you can send the moves over to a server component and it synchronizing the changes locally to the file. Of course, this means you will need to create a server component that would do something like this
user1's moves <--> server <--> user2's moves  . Server will modify the moves file.
Once you start doing this, you get into the realm of server programming / preventing race conditions etc. It will be a good learning experience.
Python over a network share