it is not recommended for people to drink distilled water due to the fact that it is acidic (we apparently need alkaline water, filtered water tends to be neutral which is acceptable) - a result of minerals being removed.
i don't know if this has anything to do with chinchillas. but, i would be a little weary using distilled water. fair enough, it doesn't contain chlorine but everything else is removed from it as well and, it's not available in the wild at all (i know our chins are not wild but you know what i mean). i think a lot of people don't understand what distilled water actually is - one of the websites i found mentions 'Water: Distilled Water. (I suggest spring water)' - this is NOT the same thing (unless you distill spring water at which point it stops being spring water)!!!
i would probably go with filtered water especially if water where you are is quite hard. or, a low residue bottled water.