Darud After The Adhan
The Shafi and Hanbali scholars are of the view that Darud after the adhan by the Muadhin should be uttered by him silently. See Muntaha al-Iradat (1/130), al-Mughni (1/428) and Mughni al-Muhtaj (1/141).
The Hanafi and Maliki scholars however regard the Darud to be recited aloud after the Adhan to be a good innovation (bidah hasanah).
Sh Ahmad al-Bashishi in his treatise called ‘Tuhfah al-Saniyyah fi Ajwibah al-Asilah al-Mardiyyah’ mentioned that that the first person to add the Salutations and Blessings upon the Prophet (Allah bless Him and give Him peace) after every Adhan upon the minaret was during the time of Sultan al-Mansur Haji Ibn al-Ashraf Shaban, which was in Shaban in the year 791.
It was introduced before this during the time of Sultan Yusuf Salah al-Din bin Ayub, and it was said before the Adhan of Fajr every night in Egypt and the Levant (al-Shaam): Assalamu Alaika Ya Rasulallah. This continued till the year 777.
There was an order to add to it by Salah al-Din al-Barlasi so that it be said: Assalatu Wassalamu Alaika Ya Rasulallah. This was then added to the end of every Adhan in the year 791. See Hashiyah Ibn Abidin (1/261) and al-Dusuqi (1/193)
Evidences and Sh Abu Bakr al-Mulla
In brief the scholars that argue that it is permissible to recite aloud the Darud after the Adhan mention that the hadith which encourages darud does not mention whether it be aloud or silently, the matter therefore has some leeway, and due to a lack of a prohibiting evidence (in their view) it is acceptable to do so.
Likewise I asked Sh Yahya al-Mulla about the loud recital of Darud after the Adhan during a lesson, he mentioned that Sh Abu Bakr al-Mulla issued a legal verdict (fatwa) mentioning that the practice was permissible. (The Sheikh used the words ‘La Bas Bihi’, and the context of the answer indicated to the practice being permitted even thought the term ‘La Bas Bihi’ can denote ‘Karahah’ in some contexts whilst at other times used to refer to something which is desirable.)