Can the Ahlulbayt and the angels read our minds and intentions? And why has Allah (SWT) allowed Satan to read our minds and intentions and misguide us?
This question touches on the nature of reality - for instance, are thoughts part of reality?
In the material world, one's thoughts and intentions are not visible or "real" in the same way that a brick or truck is real.
However, in the hereafter, one's thoughts and intention will be made manifest so everyone can see them. (Thus, everyone will be able to read what was on other people's minds the same way they would see a brick or a truck here.) Therefore, they have an objective reality which we do not see directly.
Beings such as angels or Shaytan which are occupying a different aspect of reality (for instance, coming and going from the heavens) or which are of a different construction may be able to see the realities of our thoughts and intentions. (If the angels could not see your intentions, how could they write down that you should have extra reward when you are tempted to do a wrong deed but abstain?)
To my understanding, this is not about special permission from Allah but rather being on a different "wavelength" of reality or having different senses, in the same way that some animals can hear certain frequencies that humans cannot.
In any case, the way that the universe is created is part of the divine plan, so we should assume there is divine wisdom behind it.