If my two neighbors own a cat, it’s "my neighbors’ cat". If someone famous owns a dog, what's the correct way to describe it: "someone famous’ dog"?
[It's] someone famous' dog.
This may be "correct," in the sense that someone famous is a noun phrase, and you are adding a possessive to it, but it sounds very awkward.
Even though it's a noun phrase, the second word is an adjective—and you don't normally see a possessive come immediately after an adjective.
One way of making it sound less awkward would be to reverse the adjective and noun, resulting in the possessive being put immediately after the noun instead:
[It's a] famous someone's dog.
While less awkward, it's still not the most natural sounding of sentences.
Aside from the answer already provided, you could also phrase it as:
It's a dog that belongs to someone famous.