Telepathy is a property that allows you to detect thinking creatures. Unlike most properties, there is actually a difference between when you have intrinsic telepathy (always active as a property such as by eating the right kind of corpse) and extrinsic telepathy (active from wearing a piece of equipment, like an amulet of ESP or helm of telepathy).
Intrinsic telepathy allows you to see every thinking creature on the current level if you are blind. You'll be able to see their locations and examine them just as if they were in your normal field of vision. Many people carry a blindfold to blind themselves when needed just for telepathy. One very common strategy is to blind yourself every time you enter a new dungeon level, in order to get a sneak peak at what is waiting to kill you.
Extrinsic telepathy gives you the same benefits as intrinsic, but also gives you vision of thinking creatures that are within 5 or so spaces from you, even when you aren't blind. Because it is superior, extrinsic telepathy will supercede intrinsic telepathy if you have it.
The monsters you can't see are creatures that lack minds. Put some thought into it if you'd like, but I've included a list for convenience here.
This includes jellies, puddings, fungi, lesser undead like zombies and mummies, golems, lights, vortices, and elementals.
As a warning, telepathy also gives you increased vulnerability to the mind blast attack of a mind flayer. Noted by a message of it locking into your telepathy, it can use this ranged damage attack at a much greater range if you have telepathy. So, be mindful of this.
The benefits of detection far outweigh the negatives. As such, most adventurers will seek to try and eat a corpse of floating eye as early as they can.
Best Answer
Pets cannot gain intrinsics from eating corpses. They can gain levels from eating wraith corpses, to a certain extent. They can also gain intrinsics through other methods (e.g. wand of speed).
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