What happens if your diaphragm is half paralyzed? Does it mean that you cannot breathe on your own?

Difficulty breathing. Although the diaphragm is one organ, both sides act almost independently, but they need each other for perfect breathing. A nerve on each side of the heart makes the diaphragm move. If that nerve is injured, that particular side of the diaphragm is paralyzed, making breathing difficult. This may be fixed by a surgery called diaphragm plication, performed by a thoracic surgeon.
No. Breathing is controlled by your diaphragm as well as your chest wall. Having a paralyzed diaphragm will interfere with your breathing, but you will be able to breathe.