I have created a simple chat program in python that allows many clients to connect to a single server.  I now want to create a two server model, still with many clients, so that the clients will be able to connect to either server.  Then when a client sends a message to server1 it will broadcast to all its connected clients and also send to server2, which will then broadcast to all its (server2's) connected clients.  The part I am stuck on is the server to server communication. I would like to do this on my local LAN on the same subnet; I am not trying to do this across the internet so no need to worry about dns or other protocols.
Not sure if I need to do multi-threading here or just add to the while loop that maintains connects and sends/receives the data.
Any help here would be appreciated.
I don't think that threading is an issue here. You can design a solution with or without it.
In short, your servers are not really different from your clients. They connect to other servers and send text/data to them. The only thing you have to handle specifically is the re-broadcasting of the clients chats.
This is particularly tricky and is subject to a lot of problems. You can check out how IRC handles it. You will face multiple problems which might be mitigated if you stay within a LAN. In IRC, all servers are rather equal so bringing a bunch of them down shouldn't affect the whole network (in reality this is different :p). 
What if server1 broadcasted message1 with timestamp[5 secs] then receive message2 from server2 with timestamp[2 secs]. For the client messages will appear out of order.
One thing you could do is to elect a master server for all other servers. This one will contain the master list and will manage the timestamps. All other servers will send their clients messages to it and will wait for a synchronization of the "master chat log", then broadcast the new data to all connected clients. Some messages might appear out of order for the clients if there is a lot of lag, but at least the timestamps will match and will be incremental
Distributed Local Chat Servers