The right side is not a "bad" sleeping position.
For many people, it is simply the position that feels most natural. Side sleeping in general can also be useful for people who find that sleeping on their back makes snoring worse.
People with certain respiratory problems may experience changes in breathing depending on their position, although the ideal position varies from person to person.
There is also an important distinction between right-side sleeping itself and the habits surrounding it.
For example, if someone sleeps on their right side while using an appropriately supportive pillow and mattress, they may wake feeling perfectly rested.
Another person might sleep on the same side with their neck twisted sharply, their shoulder compressed, and their spine poorly supported. That person may develop morning stiffness or shoulder discomfort.
The problem isn't necessarily the right side. It may be the sleeping setup.
A pillow should generally keep the head and neck aligned with the spine. A pillow between the knees can also reduce twisting through the hips and lower back.
If your shoulder becomes sore, slightly changing your position or using a pillow to support the upper arm may help.
Ultimately, there is no universal "perfect" sleeping position for every adult.
Chapter 5: When Should You Pay Attention to Your Sleeping Position?
If you sleep on your right side and feel healthy, breathe comfortably, and wake refreshed, there is generally no reason to panic or believe you are damaging your organs.