"Our dogs are wet."
Translation:Nasze psy są mokre.
No, grammatically speaking animals are never 'personal'.
The division in plural is between "masculine personal" or "virile" (groups including at least one man) and "not masculine-personal" (non-virile), which is for any other plural noun. Including plurals of masculine nouns (like "pies") which do not denote people. And including "women", "girls", etc. who are obviously persons but not masculine.
Does that answer your question?
Remember though that "nasi" and "mokrzy" are used if the noun in question is masculine personal. Because "pies" is masculine but not a person (because it's a dog), then in the plural we use the non masculine plural words "nasze" and "mokre."
Close. If it refers to 'a group of people with at least one man', it's masculine personal, ergo: nasi. Otherwise it's 'nasze'.
Here, a dog is both animate and masculine, but 'dogs' are still 'nasze', because dogs aren't people.
The notion of 'animate' only matters in Accusative of masculine singular nouns.