There is, I believe, such a thing as wisdom, and this is often concentrated in religious traditions. So I would expect the Dalai Lama and the Pope to have better moral intuitions than people picked at random. Perhaps I'm wrong about that, and perhaps it would be easy to find religious leaders who are bad people. But my point is really just that not everyone is equal when it comes to wisdom, and that this is relevant with regard to mortal intuitions. The wise are likely to have better intuitions than the rest of us. In that sense there are expert moral intuitions.
Possibly related to this is the fear I feel when I find out that courses on ethics are being taught by people with no philosophical training. Philosophers can certainly develop and encourage bad ways of thinking about ethics, but there are at least some pits that they are unlikely to fall into, and that can be hard for others to avoid, e.g. crude forms of relativism and subjectivism. In that sense I think philosophers do have a kind of expertise.
Following the link Leiter provided led me to this paper on human rights, which looks misguided in a way that I think most philosophers would avoid. The abstract, in part, reads:
A striking feature of contemporary human rights scholarship is the extent to which it has turned its back on the idea that human rights can grounded in a theory of human nature. Philosophers, social scientists, and political and legal theorists thus frequently assert that the classical Enlightenment project of supplying a naturalistic foundation for human rights is dead. The main purpose of this contribution to a new book of essays on human rights is to rebut this pervasive skepticism. Drawing on recent work in the cognitive science of moral judgment, I defend one of the critical premises of ancient philosophy, Enlightenment Rationalism and the modern human rights movement alike: that human beings are moral and political animals, who are endowed with a moral faculty or sense of justice. The chapter thereby seeks to offer a new perspective on an old and venerable argument about the naturalistic foundation of human rights.
This new perspective begins from the observation that whether human beings possess a common moral faculty is not primarily a philosophical, political, or theological question, but an empirical question that belongs in principle in the cognitive and brain sciences, broadly construed. The confident assertions of skeptics such as Michael Ignatieff, Richard Rorty, Gilbert Ryle, Alasdair MacIntyre, Sigmund Freud, Ruth Benedict, Richard Posner, Robert Bork, and many other writers notwithstanding, one cannot therefore simply decide the matter from the armchair. On the contrary, probative evidence and sound scientific argument must be brought to bear. [...]
... classical accounts typically rest on the claim that an innate moral faculty and with it principles of justice, fairness, empathy, and solidarity are written into the very frame of human nature. These themes were particularly influential during the Enlightenment, when the modern human rights movement first emerged. It is precisely this set of ideas that modern cognitive science, liberated from the crippling methodological restrictions of positivism, behaviorism, historicism, and other discredited theoretical frameworks, has recently begun to explicate and to a substantial extent verify. This new trend in the science of human nature, I suggest, has potentially profound implications for the theory and practice of universal human rights.One mistake that might be here (I'd have to read the whole paper to tell) is that of thinking that a common faculty of moral judgment would give human beings a special moral status. But surely having the ability to discern moral value is not the same thing as having moral value itself. So if human beings have rights it can't (simply and straightforwardly) be because we have a moral faculty.
Another apparent mistake is the idea that anything about ethics, anything normative, would follow from a discovery about human brains, such as a discovery that they have an innate sense of fairness. From "human beings have an innate tendency to x" it does not follow that x is good. What we naturally consider fair or just might not be fair or just. Mill points this out, I think, in Utilitarianism. Familiarity with that kind of insight is the kind of expertise that moral philosophers have, and that others often appear to lack.
This point might seem to be the same as the point in the paragraph before it, but it's not. Even if we have the ability to tell what really is just I don't think it follows immediately that we have any moral value ourselves. The ability to tell what is right is a good thing, of course, so we might have value just as possessors of this ability. But this value might not be very great, for instance if we never use our ability for good, or if telling what is right and what is wrong is supremely easy for every being, so that beings that have the ability to do so are nothing special. Perhaps more to the point, from the fact that a being can tell what is just it does not follow that one must not, say, beat or enslave that being.
Well, that's probably enough for a critique of a paper that I haven't read. The key mistake I think is here:
whether human beings possess a common moral faculty is not primarily a philosophical, political, or theological question, but an empirical question that belongs in principle in the cognitive and brain sciences, broadly construedThe empirical question is whether human beings possess a "moral faculty," i.e. something that scientists call a moral faculty. Whether they possess a moral faculty (i.e. whether this faculty, if it exists, actually identifies what is right) is a question for ethicists, i.e. moral philosophers, i.e. philosophers. These philosophers might well include scientists. Indeed, they include everyone who thinks about what is right and what is wrong. But doing that is not "an empirical question that belongs in principle in the cognitive and brain sciences, broadly construed."
The author of the article wants to bring together work in several disciplines, "including experimental philosophy, developmental and social psychology, cognitive neuroscience, primatology, anthropology, comparative criminal law, and other fields." Good reason to be suspicious of experimental philosophy is given by Lars Hertzberg here.