I'd like to hear your answers to this question – my own experience is doubtless atypical. Short story writers I relished as a “tween” include P.G. Wodehouse, Saki, James Thurber, Roald Dahl, Giovannino Guareschi, John Wyndham... this was 1970s England that I was growing up in... I also remember stories by Bertrand Russell and by George Mikes. And I was much taken with the mysterious short stories of Robert Graves and Graham Greene.
By my late teens, I was ready for Borges. Although come to think of it, I actually read “The Circular Ruins” long before I knew who Borges was, in a pulp anthology of horror stories. There were other amazing stories I first saw in anthologies targeted to boys, “Hope” by Villiers de l'Isle-Adam for example.
Childhood influences cannot be completely shaken off. The short story tradition is so rich, however, that there are plenty of writers working today -- writers I admire -- who haven't read many stories by the authors I just mentioned, and whose influences are quite different. This diversity is good for the field.
It was because Borges praises Kipling and Stevenson that, in my twenties, I went on to read their stories -- it's rather odd that a British writer should discover those authors courtesy of an Argentine. Chesterton's stories I'd dipped into earlier, but it was only in my twenties that I really got into him. That was also when I began seriously to read science fiction – I now believe about half of the greatest twentieth-century short stories in the English language are science fiction, but as a kid I was prejudiced against the genre. Forced to pick a best twentieth-century U.S. short story writer, I might well go with Avram Davidson.
And I think the best living short story writers in the U.S. today are Mark Helprin and Gene Wolfe. But they're both Republicans. So sue me.