The NHL changed its policy about disclosing injuries in 2008, decreeing that teams "no longer are required to disclose the specific nature of player injuries." Teams were required to announce if a player would miss a game because of an injury, and they were prohibited from providing "untruthful information" about the nature of an injury, but they wouldn't have to disclose the specifics that sports fans are used to. (Compare all that secrecy to the thoroughness of the typical NFL injury report, which are roughly the length of your average Robert Caro book.)
Teams, naturally, take full advantage of the leeway they're given by that language. But this isn't necessarily about a player's right to privacy. NHL teams don't want opponents to target injured players -- or, more specifically, the injured parts of injured players. The thinking goes that if you announce a player will miss a game because of a sore wrist, you might as well paint a bullseye on that wrist when he returns to the lineup.
In fact, if disclosing the nature of a player's injury is potentially beneficial to a team, you can bet they'll come up with something more specific that an "upper body injury." To use the Rangers again as an example, during the first round of last year's playoffs, center Brian Boyle had to leave Game 5 after a hit from Ottawa's Chris Neil. Rangers coach John Tortorella believed the hit to be dirty, and he made a point after the game of saying that Boyle had a concussion.
So why would Tortorella -- who famously gave very little to reporters during the playoffs last season, particularly after losses -- dish on the nature of an injury? Because the NHL (stupidly) considers whether an incident resulted in an injury when it doles out supplemental discipline, and Tortorella, presumably, was trying to get the league to suspend Neil. (Neil wasn't suspended, nor should he have been.) When it was in the Rangers' best interest to give out the nature of an injury, they did, but examples like that are rare. Which isn't to say teams never disclose specifics about injured players, but such things become less common as the games get more important.