As most people today are not trained in performing circumcisions, it is customary for the father of a newborn boy to hire or appoint a professional Mohel to perform the Berit for his son. The conventional understanding is that the father fulfills his Misva by appointing the Mohel as his "Shaliah" – agent – to perform the Misva on his behalf.
Surprisingly, however, there is a good deal of controversy surrounding this issue, which today is taken for granted.
The Shach (Rav Shabtai Ha’kohen, 1621-1662), in Hoshen Mishpat (382), cites the discussion of the Rosh (Rabbenu Asher Ben Yehiel, Germany-Spain, 1250-1327) in Masechet Hulin concerning the case of a Mohel who "steals" the Misva of Berit Mila from a different Mohel. The Rosh writes that if a father appointed a certain Mohel to circumcise his son, but a different Mohel then comes along and performs the circumcision, the second Mohel is not required to compensate the first. Although generally one who "steals" a Misva from his fellow must pay compensation, the Rosh writes that the first Mohel does not receive compensation in this case. The Shach raises the question of why the second Mohel is not required to compensate the father, whose Misva he "stole." After all, circumcising a child is the father’s Misva, which he was going to perform through his appointed Shali’ah, and the second Mohel denied him this Misva. Seemingly, then, the second Mohel should compensate the father.
The Shach answers by establishing, controversially, that the Misva of Berit Mila cannot be performed via a Shali’ah. If the father is unable to personally circumcise his son, then, according to the Shach, he cannot perform the Misva. The Misva falls upon the Jewish community generally, and thus anyone who wishes is entitled to come along and seize this Misva. Therefore, the second Mohel did not "steal" anybody’s Misva, and he was no less entitled to this privilege as anybody else, once the father was not performing the Misva himself. The Shach proceeds to strongly condemn the widespread practice of fathers inviting Mohalim to perform the Berit instead of performing it themselves (assuming they are able to). He insists that a father receives no credit for the Misva of Berit Mila by appointing somebody else to perform his son’s circumcision. The Shach calls upon the Rabbinical leaders of his day to urge fathers to perform the Berit themselves.
Indeed, there are some fathers who ask the Mohel to set everything up for them and show them where to cut, so they could perform the Berit themselves.
This is not, however, the generally accepted opinion. The Rama (Rav Moshe Isserles of Cracow, 1530-1572) writes clearly (Yoreh De’a 264) that the Misva of Berit Mila can be fulfilled via a Shali’ah, just like many other Misvot. Among Sephardic Halachic authorities, this is the position of the Hida (Rav Haim Yosef David Azulai, 1724-1806), in Shiyureh Beracha; the Ben Ish Hai (Rav Yosef Haim of Baghdad, 1833-1909), in Rab Pe’alim; Rav Haim Palachi (1788-1868); and the Peleh Yoetz (Rav Eliezer Papo, 1785-1828).
Therefore, the widespread practice of bringing a professional Mohel to perform a Berit is well-grounded in Halachic sources, and is certainly acceptable.
It must be emphasized that there is a general Halachic principle of "Misva Bo Yoter Mi’bi’shluho," which establishes that it is preferable to perform a Misva oneself rather than fulfill one’s obligation by appointing a Shali’ah. Therefore, if a father is capable of personally performing the Berit Mila, it would be preferable for him to perform the Misva himself rather than discharge his obligation by appointing a Mohel.
It has become common for the father to explicitly appoint the Mohel at the time of the Berit as his Shali’ah to perform the Misva on his behalf. Many people follow this practice, though it is questionable whether this is necessary, as one could reasonably claim that calling the Mohel and asking him to come to perform the Berit qualifies as the father’s formal appointment of the Mohel as his Shali’ah. Perhaps, this practice developed because sometimes a different family member – and not the father – calls the Mohel and asks him to perform the Berit, and thus the father needs to expressly appoint the Mohel as his agent in the performance of the Misva. Another reason might be that the formal appointment must be done only at the time for the Misva has arrived, and thus calling the Mohel before the eighth day does not qualify. (Although, the Hatam Sofer maintained that if the situation requires the performance of a Misva, and all that remains is for time to pass, then a Shali’ah can be appointed even before the time arrives.)
In any event, it would seem that strictly speaking, the father does not have to explicitly appoint the Mohel as his Shali’ah at the time of the Berit, but there is certainly nothing wrong with doing so, and, as mentioned, this is the common practice.
Summary: If a father knows how to perform a Berit Mila, it is preferable for him to circumcise his son, rather than appoint somebody else to fulfill this Misva in his stead. It is customary at the time of the Berit for the father to formally appoint the Mohel as his agent in the performance of this Misva.