The fly agaric is the mushroom of illustration, scarlet with white flecks, and among several peoples of Siberia it had a documented ceremonial use unlike anything in the Americas.
Among the Koryak, Chukchi and others, dried caps were eaten by shamans and by others at gatherings. The chemistry is not that of the tryptamine mushrooms at all. Ibotenic acid and muscimol act on a different receptor system entirely, and the reported effect is heavy, unstable and sleep-shot: waves of energy and lassitude, distortions of size and distance, vivid dreaming, and often severe nausea early on.
The detail that everyone remembers is well attested rather than a traveller's slander. Muscimol is excreted largely unchanged, so the urine of a person who has eaten the mushroom remains active, and it was drunk, sometimes through several people in succession. In a society where the mushroom was scarce, expensive and did not grow everywhere, this was a rational way to extend a supply, and it also removed some of the sickness of the first dose.
Reindeer are part of the picture and not merely a folk detail. They seek the mushroom out, and the association between the animal, the mushroom and the herders who depended on both is a genuine feature of the ethnography rather than a modern invention.