The beans will receive nitrogen from symbiotic organisms. They won't necessarily share much of it. Beans and other legumes are known for adding nitrogen to soil when they are tilled under to rot, not by the process of growing.

I have read several descriptions of the three sisters technique and all of them that were specific made it clear that a good amount of spacing was required between the corn plants. I think the standard corn fields I've worked in as a kid had equal or greater spacing than you describe and there wasn't much light at the ground for growing weeds, let alone squash.

The best description of 3 sisters spacing I've gotten was from Steve Solomon's book Gardening When it Counts. As I recall, he stated 4 feet between each corn plant. Now he's very anti-dense-planting but look at it like this. The corn now needs some more light on its stalk to grow the beans. The squash needs light to hit the ground between the puffy corn/bean stalks. If you plan on finding and retrieving the beans and squash you will need to see and reach through the jungle you've made.

How about trying this? Divide your bed into 2 4x8 sections. Plant one section the way you described and the other section with corn spaced 2'x2'. That should help the plants tell you what they can tolerate and still provide plenty of produce for you.

I've just noticed you posted this in may... so how about some feedback on how it turned out?