Rav Shemuel Pinchasi, in his sefer on the laws of Shabat, addresses a number of interesting cases involving telephone calls made from a place where it is not Shabat to a place where it is Shabat.
If a gentile in the United States makes a phone call on Shabat afternoon to a Jew in Israel, after Shabat had already ended in Israel, it is permissible for the Jew to receive the call and speak with the gentile. Since Shabat had already ended for the Jew, and the call was made on Shabat by a gentile, answering the call entails no prohibition. By the same token, a Jew in the United States may make a call on Friday afternoon to a gentile in Europe or somewhere else where Shabat has already begun. Since the Jew makes the call before Shabat has begun in his area, and he calls a gentile, and not a Jew, no prohibition is involved even though Shabat has begun in the area to where the call is placed.
Also, Rav Pinchasi rules that one may send a fax from the United States on Friday afternoon to Israel, even though Shabat has already begun in Israel. Conversely, a Jew in Israel may send a fax on Motzei Shabat to the United States, even though Shabat has yet to end in the U.S. In both these cases, the Jew sends the fax when it is not Shabat, and therefore he does not violate Shabat even though the fax arrives in an area where Shabat is observed. Nevertheless, Rav Pinchasi advises that one rely on this leniency only in situations of particular necessity.