I suspect that each player has a bowl of stones, and they have 1 lid that covers those bowls. After taking a turn, the player puts the lid over their own bowl so that the other player can easily see that its their turn.
"Drive by" game, to me, sounds like they've got the board set up somewhere in the house, and make moves over the course of a day (days?), rather than sitting across from each other and having dedicated time for the game.