Telepathy definitely allows the sending of thoughts and may also allow mind-reading, unless the descriptive text of the ability that grants it specifies otherwise. This is explained (albeit not explicitly stated) in the description of telepathy on page 36 of the Monster Manual:
A creature with this ability can communicate telepathically with any other creature within a certain range... that has a language. It is possible to address multiple creatures at once telepathically, although maintaining a conversation with more than one creature at a time is just as difficult as simultaneously speaking to and listening to multiple people at the same time.
That text explicitly states that the creature with the ability can communicate with (i.e.: impart information to) other creatures, so a creature with telepathy can definitely send their thoughts to non-telepathic individuals.
Whether telepathy allows a telepathic creature to pick up thoughts that non-telepathic creatures direct at it is less clear, but the text does compare telepathy to both speaking and listening, which makes me suspect it's intended to cover both.
Of course, as the description goes on to say:
Some creatures... have a limited form of telepathy, while others... have a more powerful form of the ability.
Telepathy abilities can vary significantly, so be sure to check the exact wording of the ability when it comes up in play.