Question

Why do we show left or right side dominance? Or both?

Why are certain people right handed or left handed? What about ambidextrous people? Can this dominance be changed somehow?

Answer 1

Our choice of left or right side dominance is born-in, as in hair, eye color, or stature. There may be external forces resulting in exceptions to our choice, but there will always be a dominant side.

Explanation:

We are born with a tendency to use a particular hand/arm, foot/leg, eye, ear more often than the other. The reason may be related to genetics - left handed parents with left handed children, but results vary.

Infants can influenced to some extent by their parents who may unknowingly be prompting them on a continuous basis through their interactions. Examples are waving (bye-bye) the same arm of an infant with the parent's chosen dominant hand, or turning cup handles and placing utensils in the parent's dominant direction.

Then there are teachers in early education who will try to enforce the "correct" hand to be used to perform tasks like writing or art.

There very few truly ambidextrous people. The majority are left or right side dominant, who have learned to do functions with both hands. The dominance itself will generally prevail, but it is always possible to train our body parts to accomplish new tasks.

Some people are very concerned about early recognition of left or right side dominance:
http://nspt4kids.com/therapy/the-importance-of-hand-dominance/

Others consider it to be a possible problem:
https://www.physiodc.com/right-side-dominant-do-we-have-a-problem/

Answer 2

in response to: I wonder if there used to be an evolutionary advantage.
I would like to quip that right-handedness had an advantage in slaying dinosaurs, and later, dragons, but ...

Explanation:

The phenomenon is relatively recent in world history, only observed in the last 1.5 million years. Looking at (the remains of) our ancestors from earlier on, there is no evidence of right-handedness.

Add to that, the preference for extended use of the right side is unique to humans, so animals do not support our idiosyncrasy.

There is more information on the evolution here:
http://www.bbc.com/future/story/20141215-why-are-most-of-us-right-handed