I very frequently use a combination of screen and stone to make pizzas, especially in making pizzas that are larger than my stone. I dress the pizza on the screen, bake the pizza for several minutes on the center or upper oven rack position until the top crust starts to brown and the cheeses start to bubble, and then transfer the pizza from the screen onto the pizza stone, which is preheated at high oven temperature for about an hour. The pizza remains on the stone for few minutes, or until the pizza is done. This final step increases the bottom crust browning and crisping, and since the pizza is firm and rigid at this point it doesn't matter that it is larger than the stone. Using this approach, I have made pizzas up to 18", the largest my oven can accommodate.
Some pizza operators also use a combination of screen and stone--such as the deck of an oven. The pizza is baked mostly on the screen, which is directly on the stone, and then the pizza is moved onto the stone of the deck oven for a final few minutes. This is called "decking" the pizza. Using a screen in this instance also reduces the likelihood of mishaps getting the pizzas into the oven. An additional advantage of using the screen/deck oven combination is that it allows the pizza operator to use some sugar in the dough. Otherwise, the sugar can cause the crust to brown excessively or prematurely before the rest of the pizza is done. Using the screen imposes a barrier between the pizza and stone and mitigates or prevents excessive or premature browning. I have tested this approach in my home oven and it works. Most pizza operators who use screens use them in impingement conveyor ovens. With proper control of the top and bottom bake parameters, they can often produce results that are similar to pizzas baked in deck ovens, especially crisping up the bottom crust. In a home setting, it is more difficult to get exactly comparable results, although by playing with the bottom heating element it is possible to get reasonably close.
Using a screen also allows you to bake a pizza faster than on a stone because it is only necessary to heat the oven air to the desired bake temperature, which takes only minutes to do. For a stone, you usually need to preheat the stone for about an hour. The shorter bake time using the screen can be an advantage in the summer when most people do not want to crank up the oven and stone to make pizzas. With the screen, the oven also cools off faster than when a stone is used.
I might add that screens are far cheaper than stones, or even tiles. A 16" screen costs only a few dollars and it can be used to make pizzas 16" or less in size. Ideally, one should own both a stone (or tiles) and one or more screens.