All ships have distilling plants or evaporators that inherently produce "Grade B" (distilled) water, which is perfectly safe for drinking and considerably more pure than the "Grade C" water that we normally drink when ashore. (The production of "Grade A" (demineralized) water is considerably more costly, and thus reserved for steam plants and other mechanical systems that remaining solutes in distilled water would damange.)
Princess provides bottled water aboard all ships, and even sets up "water stations" in port for the convenience of passengers going on shore excursions. IIRC, a small (1/2 liter) bottle costs $1.99 and a large (1 liter) bottle costs $2.99. You certainly can bring your own supply aboard, but most cabins have limited storage so I don't know where you would store it.
I've never had a problem with any ship water as it is distilled as you say. Instead of paying those outrageous prices for water, I just keep refilling the bottle and put it in the refrigerator to take with me off shore.
I sailed on the PP last summer and there was no problem with the water. If you don't want to take the water from the sink in your cabin, there's a tap on the Lido deck (in the coffee/tea area) that's available 24 hours. There's usually ice and lemon wedges available there too.
Thanks, to all, for the water comments. Since I have found ship's tap water to be acceptable aboard other vessels, and I take it that the PP has a plant that is just as good, I'll forget the hassel of bottled water.