Bilal would call the Adhan, then he used come to wait. When he would see that the prophet ﷺ had come out (of his house), he would pronounce the iqama. (Using translation from Abū Dāʾūd 537)
كَانَ مُؤَذِّنُ النَّبِيِّ ﷺ يُؤَذِّنُ ثُمَّ لَا يُقِيمُ يُمْهِلُ حَتَّى إِذَا رَأَى النَّبِيَّ ﷺ قَدْ خَرَجَ أَقَامَ الصَّلَاةَ