That is…
dog -> doggie
cat ->pussy
chien -> ?
chat -> ?
Petit(e) chien/chat, as far as I know.
The Spanish and Portuguese have endings that translate: ~ita in Spanish and ~ina (with the tilde) in Portuguese.
On second thought, ~ette works in French.
Third go around: kittie is chaton and puppy is chiot.
-et/-ette, to be more precise: we’d have garçon/garçonnet and fille/fillette, for example. There may be other diminutive suffixes, but I can’t think of them right now.
For the specific cases in the OP, the French equivalent of doggie might be chien-chien or ti-chien (diminutive of petit chien), and the French equivalent of pussy would be minou which, incidentally, shares with pussy the additional sense of female genitals. Chiot and chaton are just the common names for pup and kitten, they don’t have the connotation of doggie or pussy.
Doesn’t chatte have the same alternative connotation as well?
In the case of those little yappy dogs, wouldn’t it be raton? 
A mistake I made in my previous post: the diminuitive in Portuguese is not ~ino/ina with a tilde; it’s ~inho/inha (pronounced eenyo/eenya). In the case of some words, a “z” is added to the ending, i.e., the word for bread is pao; a bread roll is a paozinho. You’d think that after living there for two years, I’d remember this. :rolleyes:
The suffix -on gets a lot of mileage in French, n’est ce pas? Maybe not always as a diminuitive, though.
Yes. Though “chatte” is more common, but also more crass than minou.
As for the OP, I would say “toutou” for “doggie” and indeed “minou” for pussy.
For France, you’re right. Around here, though, toutou always means a stuffed animal toy and never a (real) dog. And chatte is almost never used as a term for the genitals, although I’m certainly aware of the meaning.
Two countries separated by a common language. 
Thankyou for your help!
The immediate cause of the question was my older daughter informing me that “Grand Chien” was not an adequate translation of her favourite toy “Big Doggie” and that I needed to find a better one 
We’ve had a few talks about the “one word doesn’t always map to one word” problem but I thought we could probably find a way round this one.
For “Big Doggie” I’d go with “Gros Pitou”. (Pitou is another diminutive term for “dog” which I’d forgotten to mention.) Alternatively, “Gros Toutou” where “toutou” works both in the French sense (“doggie”) or in the Quebec sense (stuffed toy).
I have a friend with some French-Canadian ancestry who’s cat is named that. I expect that’s correct. (just saw the cat, yesterday, that’s why I remember)