The ideas below go for any species really – dogs are just convenient.
Dogs can be communicated with via words. A person can say “Fetch”, “Roll over”, “Food”, etc, and a dog can learn to understand what those words mean. However, dogs can’t (I assume) speak human words – I’m guessing their anatomy differs in a way where they can’t say what humans say.
What if, instead of using words to issue commands, a person used symbols/images? Further, what if the symbols were designed in such a way that a dog could easily write down symbols? For example: dipping the paw in paint and swiping left means “food”. Dipping and swiping right means “fetch”.
If I was a dog, and I tried to invent a language, I would have a lot of problems. My main means of communication would probably be through eye contact, body language, and my voice. (Note that dogs use all of these frequently to communicate with their owners.) The dog wouldn’t have paper or pencil/pen, so they can’t write down any ideas they have. They don’t know how paper/pencil/etc are produced, so they can’t create communication tools. But a human could provide those tools to a dog.
If this worked, the dog could communicate its thoughts with humans. Then it would just be a matter of expanding the dog’s vocabulary, and seeing if they could learn how to make sentences. Further, if a group of dogs learned this language, maybe it would be passed onto offspring?