The story is told by an anonymous narrator and features two points of view, those of Jimmy and Bob. Although Jimmy is not identified, the fact that it is this cop's point of view is shown from the beginning of the encounter.
When about midway of a certain block the policeman suddenly slowed his walk. In the doorway of a darkened hardware store a man leaned, with an unlighted cigar in his mouth.
The policeman slows his walk because he sees the man standing in the doorway. The Summary section in the enotes Study Guide explains that Bob is being seen and heard via the cop's point of view.
The waiting man strikes a match to light the cigar he is holding, and in the brief flash of illumination, the officer notes that the person before him has "a pale, square-jawed face with keen eyes, and a little white scar near his right eyebrow." On his scarf is a large diamond, indicating great wealth.
Once it is established that the officer is noting the other man, then everything else Bob says and does is presumably seen and heard from Jimmy's point of view.
Jimmy leaves Bob standing alone in the doorway, smoking his cigar and waiting for his old friend. Then:
About twenty minutes he waited, and then a tall man in a long overcoat, with collar turned up to his ears, hurried across from the opposite side of the street. He went directly to the waiting man.
The tall man quite obviously is seen from Bob's point of view, and everything else up to the end is described from this second point of view. For example:
The man from the West stopped suddenly and released his arm.
“You're not Jimmy Wells,” he snapped. “Twenty years is a long time, but not long enough to change a man's nose from a Roman to a pug.”
After telling Bob he is under arrest, the plainclothes detective hands him a note. The text of the note must be read from Bob's point of view, since it is addressed to him and was handed to him in front of the brightly illuminated drugstore window. It tells him what the cop saw from his point of view when he was talking to Bob in front of the hardware store.
When you struck the match to light your cigar I saw it was the face of the man wanted in Chicago.
The dual points of view are adroitly handled, since both main characters are not clearly identified for the reader until the very end.
In the story "After Twenty Years" by O'Henry the view point is told by an unknown narrator. This type of narrator is referred to in the literary world as a third person limited narrator.
One never sees the narrator and he is not able to tell us what is going on when the people think. He simply tells the reader the events as they are happening and acts like the one paying witness to the conversation between the two men.
The two men in the story have been apart for twenty years and now they are coming together but only one of them knows who the other one is. The man who has returned to town does not know the policeman is his friend from the past.