The active player chooses a permanent first, followed by each other player in turn order. Each player will know what previous players have chosen when making their own choice. Then the doom counters are put on all the chosen permanents simultaneously.

None of the permanents are targeted. A player can choose a permanent with protection from artifacts, for example.

If a permanent with a doom counter isn’t destroyed by Eye of Doom’s last ability (perhaps because it has indestructible or it regenerated), the doom counter will remain on that permanent.