After praying in congregation at home, should the imam lead the supplication, or should each person make their own?

Both ways are practised; choose whichever best helps your family focus. Some families do their dhikr individually and then disperse, and some like the imam to recite a short supplication while the followers say amin. At home, the second way is often more educational because the children learn the words and meaning of the supplications. Keep the supplication short, in a language they understand, and mention the family's real concerns, such as a child's exams or a grandmother who is ill. Now and then, give a child a turn to lead the supplication. Do not let this difference in practice become a cause of tension with relatives who have different customs. See supplication after prayer.