It comes as something of a surprise to learn that, while a great many people talk about rights and verge on obsessing about them, very few people actually know what a "right" is. A right - philosophically and legally - is the power to do or not do some act or acts in relation to others. The "correlative" (that is, the other "side" of) a right is a "duty." A duty is the obligation to do or not do some act or acts in relation to others who have rights.
A "natural right" is a right that belongs to a human being simply because someone is human. Each human being is as fully human as every other human being, so each human being has the same natural rights. Traditionally, the natural rights to life, liberty, property, and "pursuit of happiness" (construed as the right to seek the "good" and develop more fully as a human being) are considered the premier natural rights. Other rights, including whatever is derived from custom, tradition, and positive law, are "derived" or "secondary" rights, and must be in conformity with our basic natural rights in order to be considered "just."
Rights are so important that they are what defines something as a "person." Something that has "natural rights" is a "natural person." Something that has rights only because someone passed a law to that effect, or because tradition or custom so defines the thing in those terms, is an "artificial person." In the United States, all human beings are "natural persons" by definition.
The Declaration of Independence, the founding document of the United States (despite recent movements to take away its special status), makes this clear: "ALL men are created equal," meaning all have the same natural rights. Nor are we allowed to play word games with the word "man" or "men" - in context it means every human being. Jefferson, a statesman and politician knew full well that not everyone enjoyed full civil rights, or even any rights at all, but that did not detract from the force of the statement. As it declared in the Constitution, the document based on the Declaration of Independence, we are adjured to work for a "more perfect union," which would not make sense if it had not already been recognized that the new country embodied some serious flaws, as Jefferson and the other Founding Fathers were well aware.
A "person" is thus a member of society. The social order (the common good) is constructed of institutions - rights - so that human beings can live together in peace and mutually beneficial cooperation. The State, a necessary institution (so much so that it is given the status of artificial person with rights delegated to it from the people) is charged with primary (but not sole) responsibility for maintaining the common good (defining the exercise of rights), and policing abuses, that is, stopping and, if necessary, punishing violations of rights.
A violation of a right is an offense, either against an individual or a group, even the whole of the common good if the effect is to disrupt the functioning of the common good in a material manner. As the human person is a "political animal" by nature, as Aristotle pointed out (that is, we naturally gather in groups and consciously structure our institutional environment), a stable social order is ordinarily necessary for our development as human beings. (You can develop as an isolated individual, but this is the exception, not the norm.)
No one can therefore have the right to offend, for an offense strikes at the very root of the social order, the free (if strictly defined and thus limited) exercise of our natural rights.
That being said, does everything that disgusts, irritates, or annoys another individual or group an "offense," technically speaking? Of course not. Many people in our egomaniacal culture believe so; being individualists, they fail to realize that, while another person exercising his or her rights might be "annoying," that's just plain too bad. Your rights are not violated simply because you don't like the fact that so-and-so makes noise, smells bad, lives in a house bigger or smaller than yours, or looks different.
There are times when someone's personal habits are annoying to such a degree that they constitute a public nuisance. Someone throwing a loud party in the middle of the night might be construed as violating another's right to enjoy his or her property or life in peace - if the mores of the local culture recognize such behavior as deviating from acceptable behavior. If, however, it has been accepted and customary on, say, New Year's Eve to shoot firearms and set off fireworks at midnight, then a judge would have to be very foolish to listen to a case in which someone claimed to have suffered an injury because he or she was kept awake at night on New Year's Eve by all that noise.
Similarly, if an ordinance exists that allows people to, for example, use power tools in the daytime, you have no legal recourse, even if your neighbor is running them all day long. By complaining and threatening your neighbor, you would be in the wrong, that is, committing an offense against the peace by attempting to restrict your neighbor's freedom to use his or her property as he or she sees fit within the limits defined by the law.
So, no. Humans do not have the right to offend, strictly speaking. They do have the right to annoy or even inconvenience others in the exercise of their natural and legal rights unless by doing so they cause harm to other individuals, groups, or the common good as a whole.