It's not necessarily that you meet them on the street. Like badasslatina mention, Turn Left is an episode where they go through what would have happened if Donna had turned left instead of right on her way to a job interview. She turns right (as in official canon), and she gets the job, starts dating this one guy, meets the Doctor, and almost gets eaten by a giant spider. She turns left, never meets the Doctor, the Doctor dies, other random shit starts happening, and Rose shows up to try and help fix it all. It's a string of events. You make a wrong turn, end up late for an appointment, have to do something else instead, some idea pops into your brain, you go do that, and somewhere along the line you meet quite a few people you never would have, should you have made the right turn.
That's kinda creepy thinking about it. In school when I'm done with PE I can cut through by the 7th grade hallway and get to class sooner, or I can go around past the band room and run into someone I'll later marry....
~Yvaine (Stardust 2007)