If you can fit the pans in the oven, you can bake more than one deep dish at a time.
I like to do the alternating pizza method, where I serve one pizza at a time, where people will try half-slices of several varieties of pizza throughout the evening, but you can certainly bake 2 or 3 pizzas at once if you can fit them in your oven.
I have one 14" pan, several 12" pans, and a few 9" wilton cake pans, which I have used in combination,
separated at two different rack levels in the oven.
Your main issue is that because your oven now has to move the heat around a few pans instead of just one,
you just need to keep an eye on your pizzas, rotate the pizzas from top rack to bottom rack, etc.
It's extremely helpful to have a few sheets of heavy duty aluminum foil to tent over the pizzas to keep them from burning while you do your oven acrobatics because pizzas closer to the top of the oven can burn before they are done baking.
You can bake with or without the stones, but I try to keep at least one stone in the oven on the bottom rack if possible for a few reasons: 1) The preheated stone will help maintain the heat in the oven while you shuffle pies; 2) I think the stone helps to bake a crispy bottom crust.
Photos from my last deep dish event:http://www.realdeepdish.com/2011/12-22-deep-dish-cember-and-year-end-food-porn/
I have a gas oven that I am generally happy with and know how to use to make tasty pizza. I also own two pizza stones, which I know how to use as well.
What I don't know how to do is to make more than one deep dish pizza in the same evening except one in the oven at a time. And given the 40 min bake time, that means I can't really have anyone over for dinner, unless someone--or everyone--is willing to eat a piece or two at a time every 40 minutes or so. With certain crowds, this would be OK, but for families with kids or for seniors, not so much. (I can probably help the situation a little by getting a 16" pan, which would allow me to feed more people at once, but it still maxes out too soon and leaves me with a multiple pies situation.)
Can I put two pizzas in the oven at the same time--one on the lower stone and one on the upper? Or do I put a bare rack above the stone(s) and put two pizzas side by side?
For other things that are baked--cookies and whatnot, most recipes say not to put more than one tray in at a time. Is deep dish pizza the same?