They don't see each other after New York. They left in different directions and it stays that way for the next few years. Occasionally, Natasha and Steve will catch up over coffee or Tony and Bruce will call each other just to tell a joke only they get. And then it's the next day and they carry on with their lives like nothing happened.
The world began to forget about the Avengers, they were yesterday's news. There were no more "Where Are They Now" specials and they started to relax, able to walk down the street without being ambushed by fans or attacked by paparazzi. Life was good those months.
It starts three years after New York. Clint was delivering files from S.H.I.E.L.D. to Stark Tower and somehow asking Tony where to put the box turns into agreeing to stay for a drink turns into going out to eat turns into waking up the next morning on Tony's couch with the main menu of "The Lord of the Rings: The Two Towers" on the TV.
A routine begins, Clint starts showing up at Stark Tower with something for Tony to sign or look at and ends up staying most of the night to watch a movie or to test out the new tech Tony creates. These nights might only be days to even weeks apart, but they both notice that they start to plan around them.
Almost a year after that first Night, Clint shows up to find Bruce sitting on the couch. The next, he invites Natasha (who only hesitates a bit before accepting) to come. And, though he isn't sure who invited him, since Tony looks surprised too, Steve shows up with a tin of popcorn in his hands.
(Getting Thor to come is a bit trickier, since none of them know how to communicate with Asgard, but once the group sets their mind to it, they devote several Nights to making plans and theories on how to make contact possible.)
And then, five years after New York, there's a knock on the door and everyone cries out when they see Thor standing there, looking oddly out of place in civilian clothes with Jane Foster at his side. Tony winks in response to Clint's confused expression (because, while they had two of the best minds of the 21st century, none of their plans really made any sense) and nods at Jane, who kisses Thor on the cheek before leaving.
"Now that we're all here," Tony begins, after everyone's calmed down, "We've got an important decision to make: The Breakfast Club or Ferris Bueller's Day Off?"