Do we have veins on both sides of our neck,or just on one side?

We have veins on both sides of the neck. It is very common to notice veins on one side of the neck more than the other and this is usually not indicative of any sort of problem.  The veins that you typically feel or see with the naked eye are the external jugular veins...these become fuller when you're lying on the side of the vein you're trying to feel.

Veins are present on both sides of the neck. Not seeing a visible vein on one side is probably related to the tilt and position of your head which may have been different when you were lying on your left side. Veins can collapse and can be difficult to see or feel especially if you're dehydrated. If your head is below the level of your heart veins in the neck are more apparent.

You are seeing the external jugular veins and less likely a huge prominent internal jugular vein. Veins have valves and it is possible that one of them has valvular insufficiency or is larger variant. 

There are veins on both sides of the neck and it is not uncommon for one side to be more prominent than the other.  This is just a normal variation.