Is it permissible to complete a voluntary Maghrib prayer of three rak'ahs aloud after a man prayed next to the worshipper with the intention of the obligatory Maghrib prayer?

You should have completed the ratibah (supererogatory prayer accompanying aFard prayer). However, by changing your intention from the ratibah of Maghrib to an absolute nafl (supererogatory prayer) of three rak'ahs, the ratibah became invalid, but your prayer as an absolute nafl was valid. Changing the intention from a specific prayer to another specific prayer, or from an absolute prayer to a specific prayer, invalidates the prayer. However, changing it from a specific prayer to an absolute prayer is permissible. Your prayer was valid as an absolute nafl, and your companion's following you in prayer was also valid. As for reciting aloud or silently in prayer, it is a simple matter, because what is prescribed is to recite aloud in night nawafil (plural of nafl) but not during the day, and the Sunnah is for nawafil prayers to be performed in sets of two rak'ahs.