So then, what did Paul mean by his ministry completing what Christ's "left incomplete?" I believe he was referring to the communication of the Gospel, the "Good News," to those who need to hear it in order to be saved. Jesus completed the efficacious work of Salvation, but unless that information is given to lost mankind, souls cannot be saved. Paul wrote, "For by grace are ye saved through faith; and that not of yourselves: it is the gift of God: not of works, lest any man should boast" (Eph. 2:8-9). Again, he wrote, "So then faith cometh by hearing, and hearing by the word of God" (Rom. 10:17).
Christ's death, burial, and resurrection on our behalf completed the work necessary for salvation, but unless a man believes that, and places his faith in Christ, he is doomed to an eternity separated from God. The Apostle John wrote: "That whosoever believeth in Him should not perish, but have eternal life. For God so loved the world, that He gave His only begotten Son, that whosoever believeth in Him should not perish, but have everlasting life. For God sent not His Son into the world to condemn the world; but that the world through Him might be saved. He that believeth on Him is not condemned: but he that believeth not is condemned already, because he hath not believed in the name of the only begotten Son of God" (Jn. 3:15-18).
All born again believers have a part of the work of Christ in reaching the lost.
So, are you doing your part in sharing the "Good News?"