It is difficult to say with any certainty precisely how many people on Earth eat primarily with their hands. The population of the Earth is roughly six and a half billion people, meaning over two billion people would need to eat with their hands for it to be a full third of the population.
We know that the vast majority of people living in India eat primarily with their hands, and the population of India is roughly one billion people. Many countries in south-east Asia (Thailand, Cambodia, Laos) also have a sizable portion of the population eating with their hands — these countries comprise another one-hundred million people. Throughout Africa many people rely on their hands as their primary utensil — Tanzanians and Ethiopians, for example, prefer this method of eating to using a knife and fork.
A fair estimate, based on known populations and the proclivities of those countries would be that somewhere between one and two billion people worldwide eat primarily with their hands. Given the dominance of chopsticks and spoons in much of Asia, and the knife and fork throughout Europe and North America, as well as through much of the developing world, it seems unlikely that a full third of the population relies only on their hands to eat.
Some sources indicate that a mere 250 million people worldwide eat exclusively with their hands, but fairly safe calculations would indicate that the actual number is substantially higher than that — even if it does fall short of the one-third mark.
In countries where eating with one's hands is the most common way to enjoy a meal, it is true almost without exception that the right hand is the only socially acceptable hand to use for eating. Since virtually all nations in which eating with one's hands is preferred are within the developing world, there are contemporary (or recent) taboos against handling many things of importance with the left hand — the hand used for cleaning one's self after using the toilet.