Do we assume that a seater die, with no crimp, is still made to remove case mouth flare, and return the brass to its proper spec (or slightly under) ???
If this is the case, then I would think the nominal .007" difference between the 9mm and .380 case mouths would be an issue.
I'm probably out of the loop, since all my pistol dies (with one exception, the .44AMP) have a built in crimp in the seater die. I even have old .45ACP die that does a roll crimp! (for the .45 Auto Rim, but also usable with ACP brass)
I would think that a 9mm seater die wouldn't "unflared" a .380 case enough, especially since the .380 is not only smaller in diameter but also shorter, and so wouldn't get the full "squeeze" a 9mm die is capable of, and a .380 die would squeeze a 9mm too much.
I once loaded .38 Super brass in .38 Special dies. The Super brass got squeezed down more than needed, and it wasn't "right" but the ammo did function and fire ok. Sometimes, there can be a difference between what's right, and what works.