He distinguishes between public education and public engagement:
Efforts that focus simply on increasing public understanding of science are not enough, because the problem is not merely a lack of scientific comprehension. In some cases, the public generally does understand scientific content in a fundamental way but still doesn't like it.
I wish that Leshner had space in the editorial to elaborate. He seems to be thinking about the debate over teaching evolution in schools, among other issues. Of course, not all scientists are working in areas that are in need of spokespeople to resolve public controversy, so his proposal for across-the-board training of graduate students in public communication is inefficient. One of his other suggestions is that academic institutions should reward people who engage in public outreach. This certainly seems true. Sometimes such efforts count as "service," which is widely considered less important than teaching and research. Arguably, though, communication with non-scientists is also a form of teaching.
I have two additional thoughts. First, when it comes to communicating about science and policy issues, what is lacking is not just the media and communications training that Leshner mentions but also training in normative reasoning and the role of science in society. Often, scientists are discouraged in their training from thinking about the normative assumptions in their research. Second, public outreach about scientific issues that affect policy can be (and is) undertaken by scholars outside of science. Philosophers are just one group that have played a key role in communicating about science and ethics with regard to biology education, the stem cell debate, species conservation, and disposal of nuclear waste.