Made for dogs
Start here. Human shampoo, including baby shampoo, is formulated for human skin, which is more acidic than a dog's. A product made for dogs should say so, and the better ones state the pH they are formulated at. If a label gives no number and no species, assume nobody measured it.
Fragrance-free
Fragrance is among the most common causes of contact irritation in cosmetics, and in pet products it is there for the owner, not the animal. A dog's sense of smell is far more sensitive than ours. Look for 'fragrance-free' or 'unscented', and note that 'natural fragrance' and essential oils are still fragrance.
Mild surfactants
The cleaning agents matter more than the botanicals. Gentle formulas tend to use mild, often sugar-derived surfactants such as coco-glucoside or decyl glucoside, sometimes combined with amino-acid based ones. Strong sulfate systems clean well but strip more. For a dog washed regularly, milder is better.
What the words do not mean
'Natural', 'hypoallergenic', 'gentle' and 'vet recommended' are not regulated claims. They tell you what the brand would like you to feel. Tea tree, citrus and other essential oils are natural and are also common irritants, and several are unsafe around cats.
A short ingredient list, published
A brand that prints its full ingredient list where you can read it before buying has nothing to hide. Short lists are easier to check and easier to trust. If a product does not fit, you can see why.
Try it once, properly
Whatever you choose, use it the way the label says, rinse thoroughly, and watch the skin over the following two days. A shampoo that leaves the coat soft and the skin calm is the right one, whatever the front of the bottle says.