I do believe in gernal animals can communiate within there species mostly using body language. Animals can communicate outside of their speicies in a similar. Look at the African planes, the Zebras, Gazelles and Wilderbeasts can all be seen grazing together. Birds are often seen on their backs picking off bugs. These animals can commuincate enough to understand that they are not a threat to each other and actually benefit from grazing together because their are more eyes to watch for preditors.

That being said, the HP world is not the Muggle world. The HP world has it's own set of unique rules. As we have seen repeatedly through the series just about anything is possible.

There is a way for animals to actually speak in the series. There is whole word used for that language... Parseltongue.

Parseltounge is the language of snakes. It is almost exclusively associated with Dark magic, and those possessing the ability to speak it ("parselmouths") are very rare. It appears to be a skill acquired not through learning, but through genetic inheritance or application of certain Dark magic. But the snake and the wizard speak to each other as seen throughout the series.

Harry is a Parselmouth. Other known parselmouths include: Salazar Slytherin and his descendants, including the Gaunts (Marvolo, Morfin, Merope), Lord Voldemort, and Herpo the Foul.

Dumbledore also seems to be able to understand Parseltongue. In HBP, when he repeats Morfin Gaunt's words "the big house over the way", which were spoken in Parseltongue. This means that either he himself is a Parselmouth or that he had previously viewed that memory with a Parselmouth who translated the conversation for him.

Fawkes does not have a lanuage associated just with Phoneixes but, he can understand Dumbledore when spoken to in English. In OoTP, Dumbledore asks Fawkes to warn him if Umbridge leaves her office. Fawkes does this by making one of his feathers appear with a bang in the office.

If these animals can communicate with wizards, I'm sure it can work with different magical species.

Also, Sirius was NOT genticly a dog. He was an Animagus. An Animagus is a witch or wizard who can turn into a particular animal at will. This ability is not innate: it must be acquired by magical means. Since there is only a handful of registered Animagus I would say it is not a common ability in the wizarding world. The most important thing to remember is that Animagi have full control over their transformations and retain all mental faculty, whereas say werewolves' transformations are involuntary and include severe changes in personality.

Sirius being a Animagus, combined with the fact that Crookshanks is NOT a cat but a Kneazle makes a world of difference.

A Kneazle is a magical cat-like creature closely enough related to cats to interbreed with them. Kneazles are considerably more intelligent than the average cat, and have a few knacks, in particular the ability to detect suspicious people (to whom they react badly). Crookshanks is half-Kneazle, which explains why he recognises Scabbers for Peter Pettigrew.

Sirius also suspected this.

Because Sirius, in his Animagus dog form and Crookshank, having Kneazle abilities, both had the same objective - get rid of Scabbers he is a bad wizard. They were united in this common goal and both had the same objective. Sirius being able to retain his sense of self while he is transformed into a dog allowed their communication to take place easier than your average Muggle cat and dog. Crookshanks would have been able to recoginze Sirius as a good wizard under the dog transformation.