Basically the Chat is a client server application. At Yahoo's main server the Chat server program is running which monitors all the actions that are taken by it's client's.
The chat server keeps track all the user's and rooms that they are logged into and create respectively.
The service of the Chat server is to broadcast the messages within the room if you enter certain rooms and join the group, the Chat server keeps track of all the room's and the user's that are inside that room. It broadcasts the messages to their respective users.
Whenever you type any message and say send, this message is basically sent to the chat server and now it is the responsibility of the server to handle the data or message.
So there is no role of you Hard disk or any local network. Everything is save on the Yahoo server.