When watching (steel) darts I noticed the scores are there immediately after each leg. Who counts them? Is it the referee? Is it the someone in the back watching through cameras? Is it a machine? Does the table have sensors?
According to the Darts Regulation Authority, there are two people responsible. There is a referee and a marker (also known as a scorer). The referee is the one who has the ultimate say in the score of a throw, and the marker has the responsibility of marking the score down. In general, it is very easy to see where the dart struck (especially for a trained referee/marker), so the scores are able to be put up very quickly.