Question

How does Perish Song determine which Pokemon faints first?

Answer 1

They both faint after the 3rd turn Perish song is used.

Both will faint after they moved on the third turn after the move is used. Their speed seems to determine who faints first.
After they both move on the last turn the faster Pokemon will go off the screen first, though it doesn't really affect anything since they both moved anyway.

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Source: Replay from showdown with a faster Pokemon, replay from showdown with a slower Pokemon

Hope I helped!

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Special thanks to Sciz for helping me test this. :]

Comments

Actually it does affect the battle if they are the last Pokemon on both sides, who faints first would determine the match.

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No it doesn't, the user of perish song will lose (I think so, not really sure, but it's definitely not based on speed). In the same way if your Pokemon faints of BB flare blitz recoil etc, the user will win.

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That's because the foe takes damage from Flare Blitz and then  the user takes recoil damage. But, as stated by Immortal, their speed determines who faints first, and the Pokemon who faints first would lose.