Do I need to screw down both sides of a composite deck board at each joist?
My experience with composite deck boards is that they flex more than similar sized solid lumber. I would not take the quick way out and instead fasten the deck boards in the manner recommended by the manufacturer. Most certainly the manufacturer directions will also have something to say about the joist spacing as well. 

Overall my take is that the more points you secure the deck boards to the joists the less flex that will be felt in the floor as you walk around on it.