Disclaimer: I'm not a marine biologist or food scientist or anything, and I've never brewed an oyster stout. So this is just speculation from a homebrewer, professional chef, and a guy who grew up in an area where the biggest event of the year was the Urbanna Oyster Festival.
From the couple of actual oyster stouts I've had, they have a slight meaty and briny character. That's the big difference in the different varieties of oysters, the brininess. I think this is the minerally character described earlier. The levels that would be added from the oysters would depend greatly on the waters they come from (more pronounced when from cooler waters is a decent enough rule of thumb). I would add the oysters and the oyster liquor (the juice that's inside the shell) at the beginning of the boil to fully extract this quality. Oysters are generally eaten raw or through quick cooking methods to prevent them from getting too chewy, not to preserve the flavor.
I don't think that shells would really add anything except maybe a little flavor from the the adductor muscle still being attached. When you add calcium carbonate (chalk) to the liquor, how do you usually add it? In powder form, and it's still pretty had to get it to dissolve.