You can in certain settings, such as academia, where it is usually considered unethical to submit one work in more than one place, or to incorporate significant portions of an earlier work into a later one without a proper citation. Whether or not it technically constitutes "plagiarism" more or less depends on whether you accept a definition of the term that limits it to stealing the ideas of others. Furthermore a certain amount of material recycling is permissible under fair use and frequently necessary. Even if "self-plagiarism" doesn't fit the definition of plagiarism, strictly speaking, it is still considered academic dishonesty and treated with like seriousness.
Recycling one's own speeches, on the other hand, is a different matter. Every public speaker has done it, for ever. (Ask a pastor!) I know of more than one public speaker who has a selection of lectures on various issues that he can present on request. It sounds like Newsweak is trying to impose journalistic ethical standards out of context.
I give the same speech more of less every time I do a Libertyseed.