I think this is as designed. When an initial talk request comes in from one of your contacts, Google don't know which screen you're paying most attention to at the time. Once you reply to the message from either the PC or mobile, further notifications in that conversation should only appear on the device that you replied from.
You can totally turn off notifications on one clienmt or the other, but that doesn't sound like what you're after?
To turn off notifications totally on your mobile, open the Talk app, press Menu then select Settings, and take the tick out of "IM Notifications: Notify in status bar when IM arrives". The Google Talk application and web app are blocked on the PC I'm using at the moment, so can't give exact directions, but there should be a similar option to disable notifications there.
See also How do Gmail (desktop) and Talk (Android) chats interact?