Is it possible to directly access third party web services using Ajax? Mostly I've seen that the website I'm visiting handles it on its server and then transfers the processed/unprocessed data to client browser. Is this always the case?
(yes, almost always)
Typically, when you're trying to accomplish accessing third party web services a proxy server is used to access those services. You can't reach external third party web services because they exist on separate domains and you run into the "Same Origin Policy"
Now.... there are methods for doing cross-domain ajax, but the service you are accessing must support it (there are restrictions on what kinds of data can be returned and how the requests are formatted due to the way cross domain ajax works)
A simple way to do this is indeed by using some sort of server-side proxy for your request. It works like this. You do the Ajax request to your own domain, lets say proxy.php. proxy.php handles your request, forwards it to the 3rd party service and returns te results. This way you don't get the cross-domain errors. You can find multiple examples of these simple proxy's by using the magic Google.