The description of the Great Old One warlock's Awakened Mind feature was originally ambiguously phrased (PHB, p. 110; emphasis mine):
You can communicate telepathically with any creature you can see within 30 feet of you.
"Communication with" a creature suggests that you might be able to communicate your thoughts to it, and that it might then be able to communicate back.
However, as briefly mentioned in Brazol's answer, rules designer Jeremy Crawford clarified in a (now-unofficial) tweet in 2014 that it was meant to be one-way:
For the Warlock Old One Pact Telepathy power. Is it one way or two way communication? I read it as one, but others don't.
The intent is for it to be one-way communication, but a DM could certainly rule that it's two-way telepathy, as in monsters.
As quoted in GuerillaNerd's answer, the July 2015 Sage Advice column (whose answers are reprinted in the Sage Advice Compendium) officially confirmed this intent:
Does the warlock’s Awakened Mind feature allow two-way telepathic communication?
The feature is intended to provide one-way communication. The warlock can use the feature to speak telepathically to a creature, but the feature doesn’t give that creature the ability to telepathically reply. In contrast, the telepathy ability that some monsters have (MM, 9) does make two-way communication possible.
Most recently, the latest printings of the PHB as of November 2018 changed the wording, even though the change isn't noted in errata.
The Great Old One warlock's Awakened Mind feature description now reads:
You can telepathically speak to any creature you can see within 30 feet of you.
Whereas the original printing might have suggested 2-way communication, the current wording is more clearly 1-way. Rather than "communicating with" a creature, it says you "speak to" a creature - indicating one-way speech rather than a dialogue. Thus, as of the November 2018 printing of the PHB, the rules now match the previously stated intent.