There is no need for uttering the intention or saying it out aloud. Rather, it is recommended not to talk - even by pronouncing the intention - during the Adhan or Iqamah. The dislike of this is stronger in the Iqamah, and even greater if one speaks after the phrase "Qad Qamatis-Salah" is recited. It is recommended to repeat the whole Iqamah if one talked during it. People's practice in pronouncing the intention originates from the fact that the intention is a condition in worship, without understanding that there is no need to pronounce it.