The judges and the organizers were the same at ancient Olympia, and were called the Hellanodikai. In the earliest days there were only two, both from Elis (the local area where Olympia was located). By the 5th century BCE, this number had grown to an official Board of Nine, and then Ten a century later (which became the fixed number thereafter). The Ten were chosen by lot from among the influential Elean families. The judging panels were not, therefore, made up of international participants from other Greek poleis. The Greeks probably thought this to be most fair, as there might otherwise have been the potential for misconduct from the judges of powerful states like Sparta, Argos, Corinth, or Athens.

In the months leading up to the games, the Hellanodikai lived at Olympia in a sort of barracks called, creatively, the Hellanodikaion. There they were trained to be experts in whatever event(s) they were to judge. In the Classical period, the Hellanodikai were not allowed to compete in the games themselves.

Technically, there was a Council of Olympia along side the Hellanodikai to whom controversial judging decisions could be made. I don't know of any such case, though. The Council had civic functions -- though not technically a city, the sprawling complex at Olympia had some of the trappings of one, including a Prytaneum (sort of a Town Hall) and a Bouleuterion (Council Hall). The Councillors were all locals. I had always assumed the Council to be "above" the Hellanodikai administratively, but now I'm not sure about that.

Source: Young, A Brief History of the Olympic Games (Blackwell 2004).

Another question:

Considering how close these races tend to be (literally .6 of a second between first and second place) how did people accurately decide who was first and who was second? Was it a spectator vote or something?

This may be a bit on the side of semantics, but I think it's pertinent to the discussion. The sort of "default" view is that the concept of a nation was not born until the French revolution.

There are scholars who argue that some ancient states met the criteria of nationhood, including Egypt and Athens - but I just wanted to bring up the fact that the concept of nationality or of the nation-state which we take for granted is far from universal-through-time, and we can't take for granted that people from different ancient Greek states would have viewed themselves as distinct nations from one another.

edit:

As this relates to judging, then, certainly you have competitors from different states, but the sort of ethnic rivalry that comes with nationality was likely absent.