The longer answer is also no.
You don't need a seperate ground, ONLY when the receptacle's tab is in full contact with the metal box.
...I have to add, in all fairness, that I see this rule broken a lot more often than I see it observed. I always thought the mounting screws served as ground, too, until Mike told me otherwise.
And it still doesn't make that much sense to me, the cross-section of those two screws must be pretty damn close, if not over, the cross-section of a 12-gauge wire? But I don't write or interpret the rules...
Sorry not to have warned you, Ian, I thought you were on plastic boxes.
... had to look it up.
12-gauge wire is 0.0808" diameter, 6-32 screw's shaft is 0.0997". Steel instead of copper, but there's 2 of them... Weird / silly rule, then, I think. Wish Mike or Pete was around to explain the rationale, but neither's been around, lately.