Street photography as a genre involves the observation and capture of people in public places. Because it's not staged it requires a keen eye - or a good dose of luck - to capture an interesting and transient moment in time. Henri Cartier-Bresson, perhaps the greatest exponent of this art, called it the "decisive moment", the fraction of a second which captured and expressed the significance of an event.
Like comedy then, timing is everything in street photography. We can't all be Cartier-Bresson, but we can keep our eyes open, our wits about us and hope for a dose of luck. You could perhaps set your camera on high speed multi-shot mode to improve your chances of getting the shot, but I think that's cheating and not faithful to the traditions of the craft.
You have to be prepared for the fact that it won't always come off. The first shot demonstrates a shot that was nearly sensational. I was watching the young man on my favorite steps and saw the girl approaching, but wasn't quick enough to adjust my zoom, framing and focus. Even so, had I hit the shutter just a second later, the juxtaposition of the boy, girl and the direction of his furtive gaze would have resulted in what I'm certain would have become an iconic image of the genre. Instead I have a shot just like the ones taken by your aunt which always had peoples heads cut off.
In the second photo, what caught my eye was the window and the way it framed the interaction of the two men. The woman inside the cafe, who was clearly not part of the conversation, gave a secondary point of interest to the scene. I got off a couple of shots, but as I was focusing on the men, I didn't notice the woman had hidden behind the magazine until I reviewed the images on my computer. I don't really know why - perhaps it's the implied indifference of the magazine reading woman or expression of the boy on the cover - but I think it's a more interesting image that the shots where she's visible. A a bit of observation, a bit of timing, a bit of luck. It all works.