Here's what I do: (1) burn off any spillage (2) scrape off any residue or raised debris (3) admire the well-used appearance of the stone and wistfully recall the great pizzas that contributed to the stains.
A pizza stone should look clean only once: when originally purchased and first removed from its box.
Allow the pizza stone to set inside of the oven when the oven is set to "cleaning cycle". This assumes an electric oven is used and I've used this method of cleaning my stone several times without any adverse effects.
But I know another person who bakes on his stone thru thick and thin, whether it's bread, pizza, or even a roast beef setting inside of a roasting pan. And he's never cleaned his stone.
There you have it, VOILA, from both sides of the spectrum!