How do researchers from different institutions get to know each other to begin a collaboration?
Go to a conference/workshop.  Ask intelligent questions in talks.  Talk to speakers afterwards.  (Have your advisor introduce you if you're shy.)  Eat lunch with them and their students.  Hang out in the hallways.  Talk about your research.  Listen to other people talk about their research.  Listen to gossip.  Throw in a few dirty jokes.  Grab some coffee, or beer.  Scribble on napkins.  Go out for dinner with the students.  Drink.  Play pool.  Try that new absinthe place around the corner.  Scribble on napkins.  Throw up in the alley. Stumble home.  Sleep.  Grab coffee.  Run into someone you met at the absinthe place.  Commiserate about your hangovers.  Talk about your research.  Scribble on napkins.  Have a good idea.  Skip the next four hours of talks.  Fly home.  Exchange drafts over email/github.  Chat on Skype.  Revise, revise, revise.  Submit to the next conference.  Repeat.

Very important: Do not skip the first 15 steps.