The most common cause of shoulder pain at night is due to the condition of rotator cuff tendinitis.
The condition rotator cuff tendinitis is caused by irritation of the shoulder tendons which cause the shoulder pain due to inflammation.
You could also have a tear in the rotator cuff as well.
A few other causes of shoulder pain at night are pinched nerves, overuse, disuse, sprain, strain, or sleeping on side.
Sometimes if you lay on your arm accidentally through the night it can cause your arm and shoulder to hurt due to the compression of the nerves in your arm and shoulder.
You could try sleeping with a body pillow to help keep your shoulders and arms from hurting or try sleeping on your other side when one side of your shoulder hurts and take some pain relieving medication and use a heating pad on the hurting shoulder.
I would also recommend telling your doctor that you're experiencing shoulder pain that happens at night as it could be something they can fix or they might be able to prescribe you some pain medicine to help get rid of the shoulder pain.
Sometimes shoulder pain could be caused by disease such as cancer or be leading to cancer so always get it checked out although most cases of shoulder pain especially if it only happens at night is caused by something not very serious.