[Raw data for what follows is available on request.  Just send me a message with your email address.]

Some figures based upon every one of my year 2008 BrainKing backgammon games (n=702) in which at least 2 rolls are saved in the system:

Out of 702 games played, the average expectation for the number of games in which opener's and responder's rolls will be identical = 39 games.
Observed number of games in which identical rolls were seen=123 games.

Roughly 3 times the expected frequency!

Out of 702 games played, the average expectation for the number of games in which responder's dice will both be different from opener's dice = 312 games.
Observed number of games in which this occurred=156 games.  (Somebody should double-check this.  It's very close if not correct, but I'm tired.)

That's exactly half of what's expected.