Asalam u Alaikum. All Praise is due to Allah. Coming to your question, the Imaam should say the Takbeeraat for the movements of the prayer aloud, so that the people behind him can hear him and follow him. If the mosque is big and the imaam’s voice cannot reach all the people who are praying, or his voice is weak because of sickness etc. or because that is his nature, some of the people praying behind him can convey the Takbeeraat [by repeating them aloud], because of the saheeh hadeeth which says that the Prophet (peace and blessings of Allaah be upon him) led the people in prayer when he was sick, and Abu Bakr (may Allaah be pleased with him) made the people hear the Takbeer. (Agreed upon).
With regard to the person who is praying behind the imaam, the Sunnah is for him to say the Takbeer softly. This also applies to his recitation of Qur’aan, Takbeeraat, saying Tasbeeh in Rukoo’ etc., the Tashahhud, the Salaam and the du’aa’, whether these words are waajib (obligatory) or naafil (supererogatory).
If someone doesn't repeat the Takbir during various stages of the prayer, his prayer will carry less reward in comparison to the one who is keen to emulate the Sunnah in this aspect.
الحمد لله أوله وآخره