The point of view in "The Adventure of the Speckled Band," as in other Sherlock Holmes stories, is first person limited. Specifically, the narrator, Dr. John Watson, is a first person witness narrator: he is telling a story which he has personally witnessed, rather than one that has been told to him by someone else. However, he is not the protagonist telling his own story but rather a character close to the main character whose function is to observe that character at work and record the details of his cases.
The structure of Watson's narration relies upon the idea that he is writing these stories for publication in The Strand. As such, we could argue that Watson's perspective is not so much limited as it is unlikely to be completely accurate; he is editing his story for a specific audience. This story, also, is said to have happened "in the early days of my association with" Sherlock Holmes, so it does not have the immediacy it might have if the narrator was recounting recent events. In considering first person points of view, we should assess how far the narrator is reliable or unreliable and how much he is likely to understand about the events he describes. In Watson's case, he is a witness to the story, but the distance of time may have blurred the true facts. He is also, for a period in this story, retelling the words of someone else: Helen Stoner.
While the frame narrative of the story is narrated by John Watson, there is also a section within this story which is narrated by Helen Stoner, as she tells her own story. Interjections from Holmes and Watson keep this narrative tethered to the frame narrative, told by Watson, but at these points, Watson becomes a narrator retelling somebody else's words.
Dr. John Watson, Sherlock Holmes's friend and assistant, narrates "The Adventure of the Speckled Band" in first person. This is the typical technique Conan Doyle adopts for the Holmes stories, although he does write a handful either in the third person or in Holmes's own voice. First person narration is generally well-suited to the mystery genre because (in contrast to, for example, third person omniscient) its perspective is necessarily limited; the narrator (usually) does not know any more than the reader does, which is handy for a genre that relies on delivering a surprise ending. This is also why Watson generally makes for a more compelling narrator than Holmes; Holmes's brilliance as a detective requires him to "know" the mystery's solution well before the ultimate reveal, which could obviously spoil the ending for readers. Watson, by contrast, is just as in the dark as Conan Doyle's readers, though he often adds to the story's suspense by observing and interpreting Holmes's behavior (e.g., "I had never seen my friend’s face so grim or his brow so dark as it was when we turned from the scene of this investigation").