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そんなことはどうでもいいよ。『例のあの人』がどんなだったか覚えてると思う? (Japanese TL)
Never mind that, do you think he remembers what You-Know-Who looks like? (original Harry Potter text)
I'm confused at seeing どんな with no noun following. It feels like it ought to be 『例のあの人』がどんな人だったか (what kind of person was You-Know-Who?), or something like that.
Is this grammatical? Is it common? When can it be done?
大辞泉 lists どんな as 形動=na-adjective, according to which どんなだった is a totally grammatical plain usage.
For the particular case, どんな人だった sounds like referring to the personality (the person was gentle or nasty or whatever) and どんなだった feels more appropriate when talking about the appearance.
In terms of idiomaticity, I can't give a clear distinction, but as you guessed どんなだった is less common (or less frequently used) than どんな＋Noun or どう.
I think the difference between 2/3 in English more or less corresponds to that in Japanese. 2 asks what he was like as a person and 3 about his condition. In some specific contexts, 彼はどんなでしたか can be used (e.g. the asker knows he is ill and asking the person who visited him how ill he was), but in most cases 彼は元気でしたか is more appropriate (just like how are you).
This predicative usage of どんな is not very common. Though it seems grammatical, どんなだった sounds to me a bit colloquial. I don’t think I have ever said it in my life.
From a very practical point of view, you could use it when the noun you would otherwise modify with どんな is one that vaguely describes what something or someone is like, such as 様子 and 感じ. You can always say those words to be on the safer side, of course.
