I don't think there's been any word on this. I'd imagine no - there would be different weapon loadouts for each character.
I'm quite sure each character has multiple dedicated weapon loadouts.
Kind of off topic but does anyone know if you have to wait until levels 5 and 7 to unlock heavies and lights for every character? Or will it just be a one time unlock, after which you can change your body type from the get go?
Once for all, if I'm not mistaken. It's the same like weapons and appearance.