Many professional-grade chess clocks are capable of randomly choosing a valid Chess960 starting position and describing it to the players. I suspect that in many tournaments, this is how it's done.
It is also possible to choose the position using other sources of randomness, such as rolling dice. Standard gaming dice sets include not only 6-sided cube dice (known as d6), but 8-sided, 10-sided, 12-sided, 20-sided, and 4-sided types as well.
A Chess960 position can be fairly chosen using 2d4 of contrasting colours (to choose the positions of the bishops), a d6 (to place the Queen on the remaining squares), and a d10 (to place the knights). The rooks and king are placed on the three remaining squares, with the king in the middle. This procedure also corresponds to the method of deriving a position from a position ID outlined here.