There are two types of cloning: reproductive cloning, which is intended to make a baby and thus a full human being, and therapeutic cloning, which means to create embryonic stem cells that can be used for therapeutic purposes, but not to implant on a woman's uterus to make a baby. Many countries have banned reproductive cloning, some also therapeutic cloning. In the US, the laws change state by state. This article is a little old but it's pretty thorough: Special Report: Cloning
I'm going to address reproductive cloning here.
In most countries, the Institutional Review Board (IRB) of research institutions such as universities, non for profits or even the industry, have to approve any and all research carried out that involves human subjects. These IRBs police all research carried out on human subjects. In the US, the IRBs include not only scientists and bioethicists, but also members of the general public.
Of course, it's not like the IRB will come to your lab and snoop around to make sure there is no human cloning going on. Once the research is or is not approved, the IRB assumes the lab is complying. As with many other things that are illegal or unethical, and require a large team of people to achieve, keeping human reproductive cloning a secret would be very, very difficult, as you'll need a large number of researchers who can keep a secret, as well as fertility doctors, physicians, the owner of the womb, etc.
Reproductive cloning is very, very difficult to achieve, it's a technical challenge in humans, and it is very expensive, it necessitates a significant infrastructure in addition to know-how. The first report of human cloning turned out to be a fraud, and brought disgrace to the researcher, Dr. Hwang from Korea, who now clones dogs (Korean Scientist’s New Project: Rebuild After Cloning Disgrace).
Of course, nothing prevents a very, very rich person in cahoots with unscrupulous researchers in a secret lab, and surrogate moms paid to keep quiet (if the cloned person is a guy, if it's a woman, she can be pregnant with her own clone) to carry out reproductive cloning. But this is no different than any other illegal or unethical activity in the world: sometimes people get away with it. I would argue that in the case of reproductive cloning, if a healthy baby is produced, someone will talk, as it would be a huge achievement to succeed at human reproductive cloning and people would want to take credit. Just like Dr. Hwang wanted so badly to be the first to do that, that he was willing to commit fraud.