Praise be to Allah, the Lord of the World; and may His blessings and peace be upon our Prophet Muhammad and upon all his Family and Companions.
According the dominant opinion among Muslim scholars, it is a collective duty to perform Azan and Iqama at mosque for the five obligatory prayers. In simple words, if some Muslims (in a given town or village) perform this action, others become exempt from sin.
It is recorded in Sahih al-Bukhari and Sahih Muslim from Malik Ibn al-Huwaireth that the Prophet (Sallallahu Alaihi wa Sallam) said: "When a time of prayer has begun, one of you should make Azan, and then, let the oldest one of you lead the prayer" .
The Azan is one of the demonstrative apparent rites of Islam, furthermore, hearing Azan in some place means it is a habit of Muslims.
As for one who prays alone, comes to mosque late or makes up the missed prayer, it is desirable for him to make Azan and Iqama.
Anyway, according to the majority of Muslim scholars, a prayer performed without Azan or Iqama is sound.
Allah knows best.