Telling others about Christ is a work. "For by grace we are saved through faith, it is the gift of God, not of works lest any man should boast " (Ephesians 2:8-9).
There was a branch of the Baptist denomination that fully accepted the idea that being saved was entirely a work of God that had no input from the ones being saved. They went so far as to say that if you are saved, you'll walk in the door of the church. But if you aren't save you won't come. They made no effort to reach the lost because they concluded that it must be God's will that they are lost. I think you can guess the logical outcome. The group died out from a lack of members.
The fault in the above argument is the assumption that Paul meant all works when he said "not of works." The flaw is easily seen by reading the next verse. "For we are His workmanship, created in Christ Jesus for good works, which God prepared beforehand that we should walk in them" (Ephesians 2:10). God has prepared works in advance that He expects of us to do. Not works which earn salvation, but good works that serve the will of God. As Paul also stated, "Therefore, my beloved, as you have always obeyed, not as in my presence only, but now much more in my absence, work out your own salvation with fear and trembling; for it is God who works in you both to will and to do for His good pleasure" (Philippians 2:12-13). Notice carefully that our obedience, our work, is connected with our salvation.
In other words, my efforts cannot ever earn salvation for me, but if I do not do the work of God I can keep myself from being saved. The works required of me are not of my own devising, but they are the works God prepared and demands of me.
Unless you understand this and accept it, you run into difficulties. Paul said we are saved by grace through faith, yet faith itself is a work of God. "Then they said to Him, "What shall we do, that we may work the works of God?" Jesus answered and said to them, "This is the work of God, that you believe in Him whom He sent."" (John 6:28-29). Does having faith earn you salvation? No! The Bible mentions several who believed, yet were lost. Among those following Jesus it was said, "As He spoke these words, many believed in Him" (John 8:30). Yet, to these same people Jesus then said, "I know that you are Abraham's descendants, but you seek to kill Me, because My word has no place in you" (John 8:37). Imagine that; people who believe in Jesus but simultaneously sought to kill him. Then among the rulers we find, "Nevertheless even among the rulers many believed in Him, but because of the Pharisees they did not confess Him, lest they should be put out of the synagogue" (John 12:42). Yet without confession we are lost. "Therefore whoever confesses Me before men, him I will also confess before My Father who is in heaven. But whoever denies Me before men, him I will also deny before My Father who is in heaven" (Matthew 10:32-33). Here are people who believed and yet denied Jesus.
No, faith alone does not earn a man salvation, but we all know that you cannot be saved without faith. "For God so loved the world that He gave His only begotten Son, that whoever believes in Him should not perish but have everlasting life" (John 3:16). A believer should not perish, but some men manage to do it anyway. James tells us one reason this happens: "For as the body without the spirit is dead, so faith without works is dead also" (James 2:26). Faith without works (obedience) is a dead faith. It is still faith, but not a faith that can lead toward salvation. Jesus said, "If you love Me, keep My commandments" (John 14:15).
Since it is God who commands us to teach the gospel (Matthew 28:19), obeying this command does not earn salvation, but disobeying God's command will lead to eternal damnation. Hence, we must do the will of God in order to be saved. Yet even then we must admit, "So likewise you, when you have done all those things which you are commanded, say, 'We are unprofitable servants. We have done what was our duty to do'" (Luke 17:10).