Well, you need to sense whether a stone is placed and what color it is. So right off the bat, you have twice the input variables or three states to consider.
If you go analog, you could use a hall effect sensor at each crossing: no field = no stone, weak field = white stone and strong field = black stone. And then make stones with two different sizes/strengths of magnets inside of them. A similar method could use photo detectors and translucent white stones (you may get interference from ambient light). You'll need 361 analog inputs, which is a pretty tall order. :o
A digital method could use tri-state logic and may require twice as many sensors.
Maybe a combination of a hall effect switch, LED and photo sensor, then put magnets in all stones: no field = no stone, field + light = white, field + dark = black.
To reduce the number of inputs required, you could scan across 19 rows of 19 inputs, since the playing surface will not change very fast compared to how fast a scan of the sensors could be completed.