There is no dedicated machine that serve only for chatting, as u know better, the chatting is a part of the internet and internet is international networking, so network bears an IP Address through which it recognize each and every computer, similarly when we login on net, and automatic IP Address is assigned to that computer which then globally connected to other computers and each and every path (cable or wireless media) is the destination of the chatting, along with Yahoo or other chatting messenger.
If u wana in detail, but I concisely discuss your question to make u better understand, other wise its a huge topic and u will be confused then.