I read this in an anthology a few years ago, however a particular plot line should help date when it was published. A female scientist enters the Institute she runs and logs on. It soon becomes clear she is in contact with her counterpart in a parallel Earth.
They've both had a failed relationship with the same love rat and are both in the same job. Earth 2 has some differences, another astronaut took 'one small step' and some major historical battles had different outcomes, most things are on a similar development track.
In both worlds the women are making fortunes from unpublished works by major authors etc. They discuss films and our world scientist mentions Casablanca, the other woman has never heard of it. She checks her database and declares it's some old turkey of a movie starring Ronald Reagan and Anne Sheridan and poorly rated.
Our scientist instantly offers to trade for the Humphrey Bogart version and explains to Earth 2 that Reagan is now Governor of California (thus dating this story). Earth 2 woman is nonplussed and queries this based on some film Reagan made with a monkey.
At this point another transmission comes over A A A A A and they realise yet another parallel Earth is making contact. There follows a computer 'handshake' and they are soon chatting with Earth 3.
They quickly establish they are all the same scientist with the same failed love affair and exchange a few Flash Gordon Earth 2 jokes and new woman asks if the others have a finished Edwin Drood novel in their worlds.
The story finishes with scientist 1 saying "Sooner or later, one of us will" implying that there would be a lot more Earths to contact