So occasionally around the place (on Meta) I see "Since you are a veteran" or "Veterans have this privilege".
For example, here.
How much rep does a user have to have to be considered a veteran?
My guess is that "veteran" has to be taken in context of the conversation. For example, in this conversation "verteran" presumedy would refer to users with both 3K + reputation and experience in reviewing close votes on SE.
Short answer - verteran implies not only reputation, but also some experience with the topic at hand.
You might also find the privileges page useful.