I looked more into this.
In ruby and sapphire, I'm almost positive that ANY shiny that's obtained with the over posted AR code will have a "Naive" nature.
Same with the other games. If you use the Shiny code, they'll have 1 specific nature all the time.
In Emerald, they have "Modest" natures.
In Fire Red and Leaf Green they have "Mild" natures.
If you're just trying to collect shiny PKMN, then it doesn't matter.
But if you're into using only VERY specific natures like I do, utilizing shinys isn't worth it if you want to train a powerhouse team. I like to use almost strictly ADAMANT natures.
I had a Shiny Red Gyarados on my team I raised from a level 7 (Gold or yellow) Magikarp. It had a pretty nice special attack and a pretty low Attack. Even moves like Surf that I used on Fire/ground/etc were surprisingly weak. Taught it secret power (which relies on the attack stat) and it did average damage even though the red Gyarados had a pretty low attack stat, which is pretty odd for a Gyarados.
Sig made by ME...haha