I keep my supplies in a small Pelican case, which is waterproof and fairly thick, so they're protected from damp and from rapid temperature changes. (Just don't leave your stuff in a hot car!) That way everything is packed and ready if I'm going to the lake or on a camping trip.
As far as privacy goes, do what you're comfortable with. Since you're car camping, I'd think the car would also be pretty private, and maybe easier than doing it in a tent.