According to your taste.

What will be kept on the shelves? Also, how deep will they be?

Are you asking for aesthetic or structural reasons? If structural, check out The Sagulator: https://www.woodbin.com/calcs/sagulator/. I usually start there to check the material and thickness relative to the span. Even if it's not 100% precise, if you're well within the recommended parameters you should be ok.

It will be MFD, 12 inches deep, 55 inches wide. The guy that’s making it, will only make the shelves 2 inches thick because he said it will eventually sag being that it’s 55 inches wide. I’m not a fan of the 2 inch look though. Is 2 inches necessary?

I'm with your cabinetmaker. Say he builds them at 1" and they sag. Now you're on the internet telling and showing everyone in the world that Jim's Cabinet Shop does crappy work. No one is reading Jim's I-told-him-so rebuttal. Were I Jim, you'd get the 2" or you'd get your deposit back. Your choice, but you ain't gettin' 1" shelves with his name on 'em.