Do Grandmasters think using algebraic notation? Is algebraic notation used just to communicate chess moves in written and spoken form, or should players strive to use it during their thinking process as well?

In Dan Heisman's excellent The Improving Chess Thinker, which examines the thought process of chess players as originally studied by Adriaan de Groot, Dan asks chess players to verbalise their thought process using algebraic notation. He includes several transcripts from beginners through to masters, including that of Max Euwe translated from descriptive notation, which looks like this:


  [A short extract from Max Euwe's analysis of the “de Groot A” position]:
  
  
  “1. Nxd5 Nxd5 2. Bxd5 Bxg5 – no, nothing then, 3. Rxc6 is a cute move but at the end of it all everything remains hanging. Something else:
  2. Bxe7 – he just takes back. 1...exd5 is very favourable (for me); he won't do that; it needn't be investigated...”


Do Grandmasters assess candidate moves by thinking “I take there, they take back with the knight” or “Nxd5 Nxd5”? Is the notation that appears in the transcripts a direct reflection of the actual thought process, or merely a vocalisation of what they're thinking in order to communicate their ideas to the researcher or coach?

Or, put another way, is it beneficial for chess players to think in algebraic notation (because it's quicker to think in “the language of chess”, perhaps?), or is it purely an aid for communication?
I wrote to Dan Heisman to get his take on the question. Here's his reply:


  “Everyone is a little different. The book is algebraic so others can
  read and understand it. In practice some just think in pictures; no
  verbalization is accompanying. Others might think more "If I go
  there..." Sometimes algebraic is helpful to clarify in your own mind
  what you really mean.”