The Holy Prophet Muhammad (sallal laahu alaihi wasallam) said that when we hear the voice of the Mu'azzin, we should repeat whatever he has said and then send Durood Shareef upon him. He said that the person who sends one Durood upon him, Allah Ta'ala will send ten mercies upon that person. (Sahih Muslim, Al-Qaulul Badi) In this Hadith Shareef, proof is found here that the Holy Prophet (sallal laahu alaihi wasallam) asked us to recite the Durood Shareef after the Azaan.
The Holy Prophet Muhammad (sallal laahu alaihi wasallam) said: "If, when performing any good deed you do not commence with the praise of Allah and send Durood upon me, then that deed is empty of blessings." (Al Jaame Sagheer) Which Muslim is there who does not consider the Azaan to be a good deed?
In fact it was Salahud�deen Ayubi (radi Allahu anhu) who gave the command of Durood Shareef to be recited before and after the Azaan. Hazrat Salahud'deen Ayubi (radi Allahu anhu) was a very great and pious King who liberated the Baitul Muqaddas. Great Muhaqiqeen and Muhaditheen made Du'as for him.
Hazrat Qaazi Iyaz (radi Allahu anhu) said that it is permissible to read Durood Shareef at the time of Azaan. He says, "And from among the occassions on which Durood should be sent, one of them is at the time of remembering him (Prophet Muhammad sallal laahu alaihi wasallam), when taking or writing his blessed name or at the time of Azaan." (Shifa Shareef) Hazrat Mulla Ali Qaari (radi Allahu anhu) in his book "Fathul Mueen" says that Durood should not only be recited before Azaan, but also before the Iqaamah. Sheikh Kabeer Bakari (radi Allahu anhu) has stated that it is desirable to recite Salaatus Salaam before the Azaan and the Iqaamah.
Thus, to recite Durood Shareef before and after the Azaan is totally permissible and those who follow this are the Ahle Sunnah Wal Jama'at.