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Not strictly necessary, a lot of South American hornos have no chimney, but having one makes cooking a lot more agreeable. The hot gasses coming out of your oven are not necessarily something you want blasting in your face.
An pizza oven will work just fine without this vent, but the vent will help to reduce the smoke and hot gasses in your face as you work the oven.
I think "fine" can be a relative concept. :-)
There is a lot of wiggle room with brick oven design and building, but I don't think I would consider building an oven without a vent and chimney. If you are going to the effort and cost of building an oven, I would do the vent and chimney. The oven draw, heat movement, and the way smoke leaves the oven are so much better.