Musicians challenging each other is a good thing. Competition is what gives people such a high standard of living today. Competition in the music industry is important too. When musicians challenge each other, they end up competing, which is why we end up with so much great music to listen to. In short, musicians should challenge each other to produce the best music they can.
A true musician will always be changing their craft. In order to keep yourself tuned you must find that certain thing that makes your stuff work. If you keep playing the same thing then you get old and tired and you want to be young and fresh even when you are old and tired.
The artistic world is an often fraught one, with one-upsmanship running rampant as each artist works to ensure that their piece is better than everyone else's. This extends to the music industry as well. I believe that this competition actually benefits everyone involved because it means that challenging another musician dares him or her to create their best work.
No, musicians should not challenge each other because music is not a sport. I know that music is a competitive field, but it is not something you can win at. Music is art. Art is something that takes you to another place and makes you think about things in one way or another. Music should bring musicians together.