Most professional photographers tend to follow the Rule of Thirds, unless they are going for a specific effect.
The Rule of Thirds generally creates the most compositional interest and aesthetic appeal. This is true across all two-dimensional disciplines of art, including film.
Obviously, this is not always the case. Each artist has to take into account whether the rule of thirds (which is only a rule) gives them the style and effect they are after, or if another compositional style would be more appropriate.
To address your specific question on the photos:
With the child, I would place him farther to the right, so that the
photograph doesn't feel so suffocating. Empty space would provide
interest and would lead the eye to the main subject (the boy).
With the girl, it really depends. Her body positioning lends itself
well to an almost dead center composition, but may look more
intriguing further to the left.
As a general rule of thumb, more empty space tends to provide more interest and focus on the main subject.
A Common Misconception About The Rule of Thirds:
A common misconception with the rule of thirds is the idea that the subject should be place in one or more of the nine quadrants created by the horizontal and vertical lines.
In actuality, the rule of thirds specifies that you place the subject at one of the four intersections that the horizontal and vertical lines create.