For those who think this is just another instance of two alpha-males going at it and being unable to coexist a la Shaq and Kobe, they’re dead wrong.
These two are actually communicating on the court, and not in the media. Additionally, neither can question the other’s work ethic. Bryant may have missed a defensive assignment, and Howard may miss free throws, but both work relentlessly on every part of their game. The mutual respect is definitely there.
Similarly, Bryant likely realizes that Howard’s criticism comes out of a desire to be a great defensive team, and out of competitiveness to win ballgames and eventually a championship. Both are something Bryant can highly relate to. He’s not used to having someone close to as passionate and competitive around him, so it’s something he should appreciate.
An Added Leader
Kobe’s last two championships came with guys he led; guys who he had the right to criticize about anything and who didn’t have the right to say anything to him other than maybe Metta World Peace, who has a Defensive Player of the Year award under his belt as well. 
Now, he has a dominant defensive big man who appears to be willing to keep him in check on the defensive end. Combined with Bryant’s continual pushing of his teammates to get better, this could be the recipe to make the team better as a whole.
Bryant is still “The Man” on this team and a living legend, but he’s now learning that there’s another vocal leader on the team not afraid to call his teammates out. After all, Howard did lead his team to the NBA Finals in 2009 (in which his Orlando Magic eventually lost to Bryant’s Lakers), so his leadership should be welcomed by Bryant, the coaches, and the rest of the team.
This can only have a positive affect on the rest of the players as well, as they now may feel a sense of open dialogue as well as a sense of accountability; as now if they make a mistake they’ll likely have two guys in their face about it.
Once the team starts winning ballgames, the so-called feud will be looked at simply as great communication and accountability.
Kobe warned Dwight of the tight ship in which he’s captain of, but perhaps he didn’t expect to get a dose of his own medicine in the process.
In either event, it should be something Bryant appreciates and respects–and I think it is, even if it appeared to be just the opposite at first.