Power is the product of force and velocity
P=(F*d)/t
Thus, there are two ways to improve power: increasing maximal force production, and increasing rate of force production. Put simply, this means increasing strength and increasing speed. You want more power, you need to get stronger and faster. You get stronger with heavy resistance training (usually powerlifting) and faster through methods like ballistics, olympic lifts and plyometrics. Ballistic exercises are especially great for fighters because something like a medicine ball throw can be done in the same plane as punching.
So you get stronger and faster, but you need to apply those physical attributes to fighting ability. This is where more sport specific exercises come in. It's also where coordination and technique come in. Stronger, faster muscles will always be outdone by the superior neurological adaptations for punching that come from actually punching. Shadowboxing helps with this, so it will improve power and help translate physical power into punching power. You still need to train with impact though. You need to actually hit something to get a sense of the distance and to learn how to deliver power effectively. However, even if you build the power to bend bags in half, it won't mean shit if you don't have the footwork to get in position to use it, the timing to catch your opponent or the hand-eye coordination to hit a moving target. You can build the most powerful missile in the world, but without an effective delivery system it's not worth shit.
The point is that there's a ton that goes into increasing power. Most important are the technical aspects. Shadowboxing will help, but you need so much more to be able to make use of those gains.