This is more a matter of style, but I feel that in the first example (“A policeman’s car has sirens.”), the car is special and therefore a psuedo-pronoun and, in the example, an acutal policecar, and would, of course, have sirens. In the second one, I feel that it's just a car a police officer just happens to own and therefore wouldn't necessarily have sirens.

Again, that's mostly a matter of style to me.