I have been thinking for a while that if you want to practice communicating with aliens, try octopi. If you can actually have a reasonable conversation with some of them, like you can with a house trained dog, then you are making some kind of progress.
I don't think dogs are a great example of an intelligent species just because we can communicate with them reasonably well. They have evolved to be able to participate in human society. With the exception of some breeds of herding dogs, they're probably stupider than (to pick one example) raccoons. But dogs possess a key trait that raccoons lack: they can understand human emotion reasonably well.
Chimp communication is complicated enough that it can probably be called conservation, but I think they're one of the only species for which that is true.
I think most animals lack a wide enough range of responses to have a conversation.