Halliday & Matthiessen (1999: 613n):
It is clear that animals such as chimpanzees and gorillas, whales and dolphins, communicate with signs in this defined sense, and it appears that these are in some way organised into sign systems. It is possible that some of these species have already moved towards a human-like, stratified form of language; but this has not yet been demonstrated, as far as we know, by any of the available evidence.An interesting case is that of domesticated cats and dogs. They communicate with signs to their human companions, but apparently not, or only very rarely, to each other. The affinity often felt between such pets and small children is not merely one of a shared material plane (they are more like each other in size) but also one of a shared semiotic plane: they share a common form of language.
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For a glimpse of a small portion of the protolanguage of one bird species, see Rainbow Lorikeet Semiosis.