I am not a fan of distilled water...I used to drink filtered tap water but after testing that and finding the pH about 6.5 I stopped drinking that too. I drink my tap water now, which I have had tested and it proves to be a good source of water and the pH is between 7-7.5. However, I do allow it to sit in a glass jug with magnesium oxide solids and that boosts the pH to about 9-9.5. This water tastes great and by allowing it to sit out much of the chlorine dissipates. Flouride is not used in my tap water.
However, most cannot trust their water source, if one must filter or distill then adding back electrolytes & other trace elements is a must. Grzbear, has a nice method of restructuring his water, take a look.