The Gemara in Masechet Shabbat (94) establishes a fundamental rule relevant to the laws of carrying in a public domain on Shabbat: "Chai Nosei Et Atzmo" – "a living organism carries itself." This means that one who carries a live animal or human being in a public domain on Shabbat does not transgress the Torah prohibition against carrying. A living organism carries his own weight, and therefore the animal or person is not viewed as being "carried" with respect to the laws of Shabbat.
At first glance, this Halacha seems difficult to understand. Even if a person carries his own weight, still, somebody who carries him in a public domain has still committed an act of carrying. There is no minimum prerequisite weight when it comes to the prohibition against carrying on Shabbat; a person can transgress this prohibition by carrying even the lightest of objects. Why, then, does a person not violate this law when he carries another human being?
Tosefot explain that the prohibition against carrying on Shabbat does not apply to carrying living organisms because no such activity was performed during the construction of the Mishkan. The Sages established the categories of forbidden activity on Shabbat on the basis of the activities performed as part of the Mishkan's construction, and therefore activities that have no precedent in the Mishkan's construction are not included in the Shabbat prohibitions. Tosefot note that the builders of the Mishkan never had any need to transport living organisms, certainly not human beings, and therefore such carrying is not included in the prohibitions of Shabbat.
Rabbi Moshe Feinstein (Russia-New York, 1895-1986), in his work Iggerot Moshe, poses a particularly novel theory to explain the Halacha of "Chai Nosei Et Atzmo." In his view, the prohibition against carrying on Shabbat applies only when it is evident that the item had been transported. When a person carries a live animal, for example, one who sees the transported animal in the new location will not necessarily realize that it had been carried; it may have reached that new location independently. The prohibition of carrying applies only to inanimate objects, which can change locations only if they are moved by a person, and their relocation unquestionably testifies to their having been transported by somebody.
The Mishna Berura (commentary to the Shulchan Aruch by Rabbi Yisrael Meir Kagan, the "Chafetz Chayim," Lithuania, 1839-1933) observes (308:154; listen to audio for precise citation) that many people ignorantly believe that it is permissible to carry children on their shoulders in a public domain on Shabbat, due to the rule of "Chai Nosei Et Atzmo." In truth, however, although one does not transgress a Torah prohibition by carrying a human being on Shabbat, the Sages enacted a provision forbidding carrying human beings on Shabbat, as a safeguard against Torah violations. Therefore, one must ensure not to carry or drag a child in a public domain on Shabbat, despite the fact that this would not transgress a Torah prohibition. Interestingly, the Mishna Berura adds that many people would insist on carrying their children in a public domain on Shabbat even were they to be informed of this prohibition. With regard to such people, the Mishna Berura rules, it is preferable not to inform them, so that they transgress this prohibition inadvertently, rather than intentionally.
Summary: Although Torah law allows carrying people on Shabbat in a public domain, this is nevertheless forbidden on the level of Rabbinic enactment.