It sometimes happens that a Berit is performed by different Mohalim, each of whom performs a different stage of the circumcision. One Mohel removes the foreskin (the stage known as "Mila"), and the other removes the membrane underneath the foreskin to expose the top of the organ (the stage known as "Peri’a").
Ordinarily, this is perfectly acceptable. The question arises, however, as to whether this is permissible when a Berit is performed on Shabbat. Maran (Rav Yosef Karo, author of the Shulhan Aruch), in his Bet Yosef, cites those who maintain that this may not be done on Shabbat, because the first Mohel does not perform a complete Misva act. Although circumcision overrides the Shabbat prohibitions (when it is performed on the child’s eighth day), this applies only when it is done in fulfillment of the Misva of Berit Mila. In the case of two Mohalim who perform the different stages of circumcision, the first Mohel removes a piece of skin without completing the Misva, and therefore, such an act does not fulfill a Misva, and thus it cannot override the Shabbat prohibitions. He thus turns out to be committing a forbidden act.
By contrast, the Rama (Rav Moshe Isserless of Cracow, 1530-1572), in Yoreh De’a (266:8), followed the opinion that permits two Mohalim to perform the Berit in this fashion on Shabbat. Moreover, it is mentioned in the commentaries to the Shulhan Aruch (the Shach and the Taz) that the custom among Ashkenazic communities was to allow two Mohalim to perform the different stages of the Berit even on Shabbat.
Sephardic practice, however, forbids performing a Berit in this fashion on Shabbat, in accordance with the view of Maran. This is the ruling of Hacham Bension Abba Shaul (Israel, 1924-1998), in his work Or Le’sion (vol. 2, p. 263).
Interestingly, however, Hacham Bension adds that it would be permissible on Shabbat for an Ashkenazic Mohel to perform the first stage of the Berit, and then a Sephardic Mohel to perform the second stage. The Ashkenazic Mohel is permitted to perform half a circumcision because this is allowed by Ashkenazic tradition, and then the Sephardic Mohel is allowed to perform the second stage because this completes the Misva of Mila, which overrides the Shabbat prohibitions.
Summary: On weekdays, it is acceptable for one Mohel to perform the first stage of circumcision (removing the foreskin) and a second Mohel to perform the second stage (removing the thin membrane underneath the foreskin). On Shabbat, however, Sephardic custom forbids such a practice, though an Ashkenazic Mohel may perform the first stage of the circumcision and a Sephardic Mohel may then perform the second stage.