Publicity comes from the French term publicité, which, in turn, links to the English word public. When someone or something receives publicity, they’re experiencing public attention. In this context, publicity could also refer to the act of being out in a public space.
Implied means that something is suggested but not explicitly expressed. Thus, the publicity in the sentence is hinted at, alluded to, but not the result of a direct confrontation.
Such, as the question notes, indicates that the privacies were previously documented. In Henry James’s long sentence, such plays the role of determiner; it shapes and determines the meaning of the privacies. Such privacies recall the "sallow prints on the walls," the "lonely magazine," and the rest of the unbecoming elements in Kate's private space.
Privacy is when one isn’t under the spotlight, the opposite of publicity. They’re not being looked at or talked about by the general public. They can live their life free from the rhetoric or prying eyes of other people.
Perhaps James turned privacy into privacies to emphasize the plurality of factors that make up Kate Croy’s disenchanting realm. Privacy can be read as a plural noun. James could have used privacy, but privacies arguably give the sentence a dramatic flair.
While the vulgar street doesn’t directly draw attention (publicity) to the crassness of her room, the fact that the street is vulgar strongly hints at (implies) that her private room and all that it composes (such privacies) is vulgar as well.
If this part of the sentence remains somewhat hard to fathom, consider why James might have purposefully crafted a thought that’s not easy to pin down. As a creative writer, it’s possible that James is more interested in playing around with words and sentence structure than in composing a sentence—or a part of a sentence—that can be understood without much effort.