Well, I see three different fields currently in EZP that could be involved - the "weight" and "shipping" in Product Edit, and the "Handling" in the Sales window. It looks like the "Handling" is just added manually during a sale.
It seems like the "shipping" is most likely what I've paid to have the item shipped to me (though I can't figure out how to get that to work, either). It seems that the "weight" field could really be used only for calculating a shipping charge to someone purchasing the item.
So, if there would be some way that the total weight (I don't think individual weights being visible would be necessary) could show up at the time of transaction (and prior to payment), we'd have a relatively simple way of determining the shipping charge. We have a nice "calculator" now that covers both UPS and USPS, and just have to enter zip and weight to get a solid estimate. Then, as we do currently, we can have a Product for shipping that we use for the charge.
We'd need a way of bulk-entering the weights. The way we can currently change an entire column in Product Edit is fantastic, and they'd probably need to be tweaked on occasion (the new "dimensional weight" system has a few variables that may not be evident at first).
Of course, some kind of system where the zip code is automatically tied to the weights and automatically spits out a shipping charge would be extremely nice, but with the way charges vary between shipper and change over time (not to mention the complexity of programming it, I'd imagine), it's hard to see that as being feasible.
