We all have some level of self-awareness about our own thoughts. On a minimal level, this simply makes us conscious beings.
"I posit two theorems: (1) The mind is all the information in the brain. (2) Consciousness is the brain's awareness of some of that information." - J. Allan Hobson.
Beyond that, we face difficulties in maintaining the awareness in a deep way and on an ongoing basis.
"It occurred to me: awareness no more permitted its own description than life allowed you a seat at your own funeral. Awareness trapped itself inside itself." - Richard Powers
"In observing the operation of his own mind, incidentally, Galton was faced with the 'difficulty of keeping watch without embarrassing the freedom of its action'." - Douwe Draaisma
If we persist, we may be called "intellectuals."
"An intellectual is someone whose mind watches itself." - Albert Camus
"Tetlock found that the most important difference between fox thinking and hedgehog thinking is that the fox thinker is more likely to study his own decision-making process. In other words, he thinks about how he thinks..." - Jonah Lehrer
J. Allan Hobson. The Chemistry of Conscious States: How the Brain Changes its Mind. Boston: Little, Brown, and Co, 1994. p 203.
Richard Powers. Galatea 2.2. New York: Farrar Straus Giroux, 1995. p. 217.
Douwe Draaisma. Why Life Speeds Up As You Get Older: How Memory Shapes Our Past. (2001) Translated by Arnold and Erica Pomerans in 2004. Cambridge University Press, 2005. p 3.
Albert Camus, quoted in Forbes.com, quoted in The Week, August 10, 2012, p. 17.
Jonah Lehrer. How We Decide. New York: Houghton Mifflin Harcourt, 2009. p. 242. See Tetlock, Expert Political Judgment, Princeton 2006. Based on Isaiah Berlin's classic essay, "The Hedgehog and the Fox."