I read this story in an old (50s or 60s, possibly 70s) Analog, F&SF, If, or one of the other magazines. The cop hunting the alien kept trying different chemical elements as bullets, because the alien could leave the dying body of its host if the wrong (ie, non-lethal to the alien) element was used. It may have been set in Russia (though I may be conflating another story,) with the cop referring to car models as "Zim" or "Zis", with the "m" in the former referring to Mussolini and the "s" in the latter to Stalin. Nobody believed the cop's story about the alien.
This is "Ecdysiac" (1963) by Robert Presslie. It was published in New Worlds Science Fiction, January 1963 and reprinted in International Science Fiction, November 1967.
The story opens in Warsaw where Richard Pike is waiting with for someone. Today he has prepared a special bullet for his gun.
The bullet was a very special instrument of death. Inside the wax there was a core of metallic sodium. If the wax melted, the sodium would be exposed to the air and in spontaneous combustion with the water vapor and oxygen in the air it would burst into flame.
Richard is trying to kill a man - he calls an "ecdysiac" - that he has been hunting for a while.
Pike hesitated for one second to make sure he had the right man in the sights of his gun. One second was sufficient.
This was the victim.
This was the man he had killed three times already.
He was almost within arm's reach when Pike pulled the trigger, sure that he could not miss, sure that even the ecdysiac would not have time to escape in the split-second flight of the bullet.
He mentions the types of the cars:
"I am a newspaper man," he said. "I don't need to look. I've seen and heard the same things many times in many places. In Germany they drive in a black Mercedes and the sign says POLIZEI. Here in Warsaw the car is probably a Zim or a Zis, and the sign says: MILICJA."
"You would think," he said "that the Russians would change the names of their cars. The M in Zim and the S in Zis stand for Molotov and Stalin, both of whom are now out of favor. You would think they would change the name of the factories and call the cars Ziks."
The story can be read at the Internet Archive.