めいめい is a casual, colloquial way to say 'each' or 'everyone individually'. 己がじし is far more formal and rarely used in speech.
Etymology
From classical Japanese 己 (ono, 'self') + が (possessive particle) + 自 (ji, 'self') + し (adverbial particle). The exact historical derivation is uncertain, but it is a fixed literary expression.
