¡Mira! ¡Qué diestro es!
 

Natives have said to me that they don't just say the sentence "¡Mira! ¡Qué hábil es!" and that they always say in what particular thing that person is skillful, like: "Él es hábil con las manos", etc. My question is that if the same rule applies for "diestro" like for "hábil"? So instead of saying only ¡Mira! ¡Qué diestro es!, should we say ¡Mira! ¡Qué diestro es con las manos! ???

"¡Mira qué hábil/diestro es!", for me, it's the same sentence, and I'd give the same treatment to both adjectives.
Both sentences talk to someone about a person that is skillful. Since the person you're talking to knows what the other person is doing skillfully (you're using "mira"), you don't really need to add what their hands are doing, but if you add "con las manos", it's fine as a more accurate description. :)
You may also say: "¡Mira, qué destreza/habilidad (con las manos)!"

Thank you Angelica for your kind answer. :) It is such a pleasure to be part of this forum. So if I get it right if there is a context no additional info is needed, but if there is no context you add the additional info like: "...con las manos"?
Best wishes y un abrazo

We always want to be as clear as possible, so giving someone a context to know what we're talking about is always better than expecting them to guess. ;)