Flask-SocketIO 2.0 can (maybe) do what you want. This is explained in the Using Multiple Workers section of the documentation.
Basically, the servers are configured to connect to a shared message queue service (redis, for example), and then a load balancer in front of them assigns clients to any of the servers in the pool using sticky sessions. Broadcasting operations are coordinated automatically among the servers by passing messages on the queue.
As an additional feature, if you use this set up, you can have any process connect to the message queue to post messages for clients, so for example, you can emit events to clients from a worker or other auxiliary process that is not a SocketIO server.
From your question it is unclear if you were looking to implement something like this, or if you wanted to have the servers communicate for a different reason. Sending of custom messages on the queue is currently not supported, but your question gave me the idea, this might be useful for some scenarios.
As far as using a SocketIO client as in the question you referenced, that shouud also work. You can use this Python package: https://pypi.python.org/pypi/socketIO-client. If you go this route, you can have a server be a client and receive events or join rooms.