The Shaafi'i and Hanbali opinion
The person who gives the adhaan should say the 'iQaamah. This is the view of the Shaafi'i and Hanbali schools of thought. They based their opinion on the following weak narration.
It was narrated that Ziyad bin Harith As-Suda'i said:
"I was with the Messenger of Allah on a journey, and he commanded me to call the Adhan. Bilal wanted to call the Iqamah, but the Messenger of Allah said: 'The brother of Suda' called the Adhan, and the one who calls the Adhan is the one who calls the Iqamah.'"
[Sunan Ibn Majah 717]
Their second form of evidence for this view of theirs is as follows: The adhaan and 'iQaamah are both forms of dhikr which precede the prayer. One person should handle the responsibility for both acts just like one person handles the responsibility for giving the two khutbah [before Friday prayer].
The scholars of the Hanafi school agree with the Shaafi'i and Hanbali scholars in this matter in the case where the person who gives the adhaan would feel hurt if someone else said the 'iQaamah because it is not permissible to hurt another Muslim's feelings.
The Maaliki opinion
One person can give the adhaan; another person can say the 'iQaamah. They based their opinion on the following weak hadith:
Narrated Abdullah ibn Zayd:
The Prophet (sallallaahu alayhi wa sallam) intended to do many things for calling (the people) to prayer, but he did not do any of them. Then Abdullah ibn Zayd was taught in a dream how to pronounce the call to prayer. He came to the Prophet (sallallaahu alayhi wa sallam) and informed him. He said: Teach it to Bilal. He then taught him, and Bilal made a call to prayer. Abdullah said: I saw it in a dream and I wished to pronounce it, but he (the Prophet) said: You should pronounce iqamah.
[Sunan Abi Dawud Hadith No. 512]
Their other evidence is that the goal is served. It is as if both undertook the responsibility of the adhaan and 'iQaamah together.
The Hanafi position
They support the position of the Maaliki school provided that the feelings of the one who gives the adhaan is not hurt if someone else says the 'iQaamah.
Kuwaiti Encyclopedia of fiQh