The dialogue demonstrates each man’s personality traits, because we learn about their past through what they say, and we learn what kind of man they are now.
Bob and Jimmy were friends when they were younger, and agreed to meet in twenty years no matter what was going in their lives at that time. They get a big surprise when they do meet, because Bob and Jimmy have taken quite different career paths.
Jimmy became a cop, and we can tell from the behavior he exhibits in the beginning of the story that he takes his job very seriously. Yet we can also tell by the way that he keeps the appointment that he takes his friendship seriously. For the example of dialogue, I chose a sample of Bob’s dialogue about Jimmy. Remember that Bob does not know that Jimmy is a cop. Jimmy does not know that Bob is a criminal until this meeting. When Bob lights a match, Jimmy recognizes that he is “the face of the man wanted in Chicago” and knows that his old friend is in trouble with the law. Yet we know that Jimmy and Bob were close. Look at what Bob says.
"Twenty years ago to-night," said the man, "I dined here at 'Big Joe' Brady's with Jimmy Wells, my best chum, and the finest chap in the world. He and I were raised here in New York, just like two brothers, together…"
From this quote we can tell a couple of things. We can tell about Bob that he values friendship. We can also tell that he does not know that he is talking to Jimmy, and the he does not know that Jimmy is a cop. Maybe that means he is not very intelligent or astute, I guess you could say. Either way, Bob does not seem to be the one who is on top of things in this situation. Jimmy goes and gets another cop, because the friendship means more to him even after twenty years, but he will not drop his duty. He still has to arrest Bob.