Large concentrations of the 2 groups living in close proximity to each other was something that was not very common up untill the creation of the State of Israel. Such communities were once to be found in Amsterdam, Hamburg, Germany and parts of Italy and England. There were also smaller Sephardic communties in Eastern Europe that eventually assimilated within the dominant Ashkenazi communties in Transylvania for example (Particularly Dej, Carei and Cluj) and in Poland. Romania actually managed to keep intact a separate Sephardic community up untill the Holocaust (see Yaakov Geller, Hasefardim asher b'Romania).
There was also an opposite phenomenon of Ashkenazim who emigrated to Spain and assimilated among the Iberian Jews. Some examples include the children of the Rosh (Rabbenu Asher) and Rabbi Vidal, a descendant of Rashi who left France for North Africa. It is also interesting to note that the common Sephardic surname "Ashkenazy" denotes an Ashkenazic ancestry. (More on this cross-pollination in a different post).
In the United States the Sephardim-who were the first Jews to settle these shores- replicated their parochial existence but by the 18th century there was a fairly high rate of intermarriage between Sephardim and Ashkenazim. According to one historical source (I have seen conflicting information on this) Emma Lazarus' was half-Ashkenazy as her mother was German-Jewish.
Rabbi Marc Angel in his book Remnant Of Israel ; A Portrait of America's First Jewish Congregation writes:
"Almost all of the 23 founders of of Shearith Israel were of Sephardic descent..The Sephardim seem to have constituted a decisive majority at least untill 1700...According to the 1721/1722 synagogue record book the cong. then had 37 members. Of these 15 were of Sephardic background, while 22 were of Ashkenazy background..In communities such as Amsterdam and London, relations between Sephardim and Ashkenazim were less than cordial. Each group maintained seperate congregations and communal institutions. Fortunately the antagonisms between the 2 groups in the old world were overcome in the new world. Sephardim and Ashkenazim in New York worked together, married each other, prayed in the same synagogue, and were buried in the same cemetery one next to the other." (pp. 52-23)
In Eretz Israel there seems to have been less friction and closer ties between the 2 communities at least untill the major aliyot from Eastern Europe in the 19th and 20th centuries. One exampe that indicates the cordial relations between the 2 communties is the personage of Rishon Lezion (Sephardic Chief Rabbi) Rabbi Yaakov Elyashar (Also known as the Yisa Bracha after his most famous work). Born to an old Jerusalemite Ashkenazic family, he studied under the Sephardic sages and was eventually appointed their leader.
The great Kabbalist, Rabbi Yitzchak Luria Ashkenazi (known as the
Arizal) who was the son of an Ashkenazy father and a Sephardic mother may also be an indication of this trend.
Friction and cooperation between the sides.The case of Benedict Spinoza and Uriel Acosta in Amsterdam were one of the few instances where the the Sephardic and Ashkenazy
kehillot cooperated with each other but the general attitude was one of mutual wariness and often open hostility. There was-for example- a strict
takana (ordinance) in the Amsterdam Sephardic community against patronizing Ashkenazy merchants for example (see
Benjamin Disraeli and the myth of Sephardi superiority Todd M. Endelman, Jewish History Volume 10, Number 2 / September, 1996). So we see that there is a long history of friction as well as cooperation between the 2 streams of Judaism.