The benefit of distilled in addition to RO water is that it is almost pure H2O. No germs, no chlorine, no lead, no pesticides — you name it! So this is great. Although, at the same time, both kinds of water lack naturally occurring, healthful minerals. This may result in low electrolyte blood levels on your dog and lead to over-hydration or water toxicity, among other ailments. We also read a report about puppies developing heart problems due to potassium deficiency, presumably because they were given distilled water exclusively.
It might also be that your dog just doesn’t like the flat taste of RO/distilled water, preventing proper hydration.
On the other side, highly pure water can help treat urinary problems such as infections, stones, and crystals.
Bottom line: Some veterinarians do not recommend it distilled respectively reverse osmosis water for dogs, many others do.
The question really is, does your pet get all the salts and minerals it needs from its food? In that case, distilled/RO water may be just fine. And there’s definitely no need to worry about giving small amounts here and there.
One may also ask, why do you need to serve your puppy distilled/RO water? If there’s a particular contaminant that you are concerned about, maybe there’s a better way to get it removed.
Unless your vet has told you to use distilled/RO water, steer clear of it, in our humble opinion. If you insist on using it, think about remineralization. This will let you bypass all of the potentially harmful contaminants and ensure that your dog gets all the minerals and salts it needs.