Java constructor should use setter?
If you have setter methods (setName, setFullName) in your class, then your constructor should call the setter methods.

setName(personJsonObject.getString("name"));


If you don't have setter methods in your class (your class is immutable), then your constructor should perform the evaluation.

this.name = personJsonObject.getString("name");


You should only pass to a method what it needs.  In this case, your setName method takes a String as a parameter, which is what the method needs.  If your method needs 3 or more values from the personJsonObject (arbitrary rule of mine), then you would pass the personJsonObject to the method.