We do it on the grill, either directly on the grill grate or on the "tombstone" pizza kettle set up for the Weber, or in the Big Green Egg either on the grate or on the stone, just depends on mood and what kind of pizzas we are doing. All of them get wood and charcoal both used so they have that wonderful wood-fired oven taste.
For the screens I see some of you using, we kind of keep the TV on for "background noise" and usually on cooking shows. I saw where someplace used the screens out of stovetop exhaust fans for their pizza screens. I'm not sure if they used them for actual cooking or just for serving, but they would make a good serving rack for pizza in that they would keep the bottom crisp, and they are pretty cheap.