It appears that zombies will not drop any items other than their basic drops when killed by the player.
They drop rotten flesh, carrot, potato, and iron ingot (and experience). They do not drop any armor or held items. I have tested this with a spawner on Hard difficulty in Survival mode (see attached screenshot).
I have tried a reset, and also a full uninstall/reinstall. I am not running any mods or packs or anything that isn't completely vanilla.
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Could be. I got a Looting 1 sword and used up about 3/4 of its health before a single item dropped.
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I used up all the rest of that looting 1 sword and had no other item drops whatsoever. Something is clearly broken with the probability mechanic. I haven't gotten any weapon/armor drops from skeletons I've killed elsewhere, either.
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Armor and weapon drops are still not happening without enchanted weapons. Any word on when this might be fixed, @@unknown?
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Can probably mark this as resolved. Looks like they now have normal-probability drops, since the 1.13.0 update.
Maybe try looting sword and let me know the result? It may be bad randomization.