Matt 28:19-20 is a command for all disciples (in effect) to teach the gospel. The apostles were so commanded and then told to teach their converts (us) to observe the same thing (among other things). Since it is a command, it is required.
I Pet 3:15 requires us to teach the lost when any opportunity presents itself. Giving an answer for the hope that is within us is not optional.
Approved examples are binding (required to be followed – Phil 4:9, I Cor 11:1, I Cor 4:16, Phil 3:17, II Thess 3:9, Heb 6:12, I Thess 1:6), else we don’t have to eat the Lord’s Supper every first day of the week. Since that is so and since Acts 20:20 is an approved example of teaching house to house, it is required that we practice what it entails.
The same is true of Acts 8:4. The whole church (every Christian – man, woman, and youth, verse 1) went “preaching the word.” So we must do the same today. We are to follow the Bible and following the Bible would include following Acts 8:4.
Heb 5:12 says there comes a time in every Christian’s life when he/she “ought” be a teacher. The word “ought” implies a requirement, not something optional.
Jude verse 3 commands all the “sanctified” (verse 1) to contend for the faith. Since it is a command (not a recommendation) to contend for the faith, it would be required. And since the sanctified would be talking about all Christians, that would mean trying to teach the lost is required of all Christians, right?
II Cor 5:11 implies that anybody who believes in the terror of the Lord will try to persuade men to avoid that terror. So not spending time trying to reach the lost means a person really doesn’t believe in the terror of the Lord (if I correctly understand the implications of the verse).
Conclusion: Christians should do “personal work” because we love God, we love the souls of others, and because it is required of God to do so. I can’t think of a more important work. Gospel preachers wouldn’t think of making the Lord’s Supper or prayer optional; why would we make the “great commission” optional?