This is interesting. You need to see your doctor about a few possibilities. One could be heart failure and there might be accumulation of water on one side of the chest or in the tummy which might be driving the cough. Another possibility is bronchial irritation for some reason on one side of the chest rather than another, and lastly it may be potentially due to reflux of gastric contents up the food pipe which is prone to varying with position. If it's bothering you, you should at least get a chest x-ray to have it checked out.
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