Thirteen people are confirmed dead and dozens remain missing after the TransAsia Airways ATR-72, (which was) carrying 58 people, veered out of control en route from Taipei to Kinmen, (which is) off the coast of the Chinese province of Xiamen.
"Which was" in this case is optional. Some might say it is implied. I would prefer to replace it with "while" (or at least an implied "while"). Doing so would understand the emphasis as on the number of people on board as the plane veered out of control.
As for "which is," there is a dilemma. Does "off the coast of the Chinese province of Xiamen" describe the island of Kinmen, or does it instead describe the place where (en route to Kinmen) the plane veered out of control? Both seem to be possible.
If "off the coast..." is meant to describe the island, "which is" could be inserted here. Even then, it could be implied.
If the place of the crash is what the writer intended to describe by "off the coast," "which is" would be impossible. He/she could rewrite the sentence, stating, "...veered out of control off the coast of the Chinese province of Xiamen, en route from Taipei to Kinmen."
Unfortunately, the writer neither included the words "which is," nor structured the sentence in this unambiguous way. Therefore, I am unable to determine which meaning the text is supposed to convey.
Obviously the writer of this news article did not expect his/her grammar to be deconstructed, and so unless the context reveals authorial intent, we may never know.