A friend and I, both American native-English speakers who have learned to read Arabic for our own pleasure, have read about two-thirds of the entire 1001 Nights in Arabic over the last five or six years.
I know of no dual language version of 1001 Nights. However, our 1999 Arabic edition (which happens to be from Dar Sader Publishers, Beirut and purchased at an Arabic bookstore in Los Angeles about which I have no other details) and our translation (by Malcolm C. Lyons, Penguin, set of 3 volumes of about 800-900 pp. each, made from the Calcutta II manuscript, which must also be the one we have in Arabic) have dovetailed almost perfectly. There are almost no differences between the two.