First, I would recommend you create the field per-product, not per-node. Next, you would need to make the shipping module would use the shipping field's value for your shipping costs.

I say it in that way because that's the end result we're after, but I'm not familiar enough with the shipping module to offer much help beyond that. It's possible there is a hook_ function that would help you intervene and apply the product's own shipping value. It's also possible you could do this in a rule, but it wouldn't be that simple.