It's pragmatic to say my Dog is instead of The dog of me is or I have a dog that is, because it avoids competing syntactic positions, e.g. the two finite verbs in "The dog I own is" or the disagreement between the person of pronoun and verb "[The dog of] me is". The topic can become the subject, e.g. mine, which even shows noun morphology in German mein Hund ~ meiner.
We also see "Good luck with [...]" omitting the subject and verb, You have.
With the firefighters it's readily obvious that the fire fighters extinguished is ambiguous to the firefighters extinguished _. In effect, your variant shows an unusual elision of the fully qualified sentence "The fire, that the firefighters fought, was extinguished, when the firefighters extinguished it". In German subordinate clauses word order is regularly inverted, ending with the verb, "... als die Feuerwehr das Feuer löschte". While this order also takes possesive pronouns als die Feuerwehr Ihr Feuer "their fire", it is proscribed vehemently to say der/dem Hund sein Herrchen "
NOM/DAT the dog
Gen. its owner". Instead des Hundes Herrchen, or Hundebesitzer for short, is preferred, "
GEN the dog's
NOM owner" or dog owner. The genitive -s is rare in German (frequently replaced with a dativ construction der Besitzer von dem Hund), and in English easily confused with the plural; The German plural feminine genitive is likewise ambiguous over nominative, die Frauen, der Frauen, thus constructions like Der Katzenbesitzer "the cat owner, the cats owner, the cat's owner, the cats' owner" are opaque.
Consequently it seems informative to assume that firefighter is a backformation, from a possessive marker 's that was interpreted as plural and dropped by analogy to singular inflection. Hence, the firefighters extinguished shows topicallization, and your the fire, the fire, fighters extinguished shows a proscribed redundancy. Since German likewise shows no binding morpheme in Feuerwehrmänner, this phenomenon in compound nouns may go back at least to Proto-Germanic, if it is not a latter parallel development.
This answer is focused on a very specific example, because topicallization is usually constrained to the subject, in English.