I am always impressed by the introduction of the epistle of John, especially as regards its evangelistic methodology. There a few things to notice here, some of which I have already commented upon in the past but which are worth considering once more.
In the first place we observe a repeated emphasis upon the personal experience of the author and his group (namely the apostles) with Jesus himself. Over and over again the author mentions what he has heard, what he has seen with his eyes, what he has touched with his hands, and the reality of his present fellowship with the Father and the Son. He twice mentions what he has heard (ὃ ἀκηκόαμεν, vv. 1, 3) and three times what he has seen (ὃ ἑωράκαμεν, vv. 1-3). It seems to me this has at least a double significance.
The first thing to note that is that the apostolic message regarding Jesus Christ, the message of life (τοῦ λόγου τῆς ζωῆς, v. 1), is a message about what they themselves have experienced and continue to experience. Not only does the apostle make use of the past tense in describing things seen with Jesus in the past, but he also uses the present when he says that Our fellowship is with the Father and the Son (the implied εἰμί, ἡ κοινωνία δὲ ἡ ἡμετέρα μετὰ τοῦ πατρὸς καὶ μετὰ τοῦ υἱοῦ αὐτοῦ Ἰησοῦ Χριστοῦ, v. 3). In other words the apostles have had and continue to have real, lived fellowship with God and with his Son. This is the reason why they dare to tell other people about it, because it is something real for them. The apostles did not make up stories about Jesus, and they did not dedicate their lives to an unattainable but inspiring ideal with no basis in reality. Their appeal is always to the empirical: what we've touched, what we've seen, what we've beheld with our eyes, what we've heard with our ears. This is why Jesus mythicism has no basis in reality, and why convinced mythicists are the most lamentable and miserable people on the planet: there is absolutely no basis for what they say in the biblical texts.
I further gather from this passage that the purported evangelist must herself have experience with the Father and the Son (and the Holy Spirit). What's the use in telling people about a reality you haven't experienced yourself? What's so good about evangelizing persons to bring them into fellowship with the Father and the Son, if you yourself do not live in this fellowship and experience it? Why should anyone believe you? To my mind, evangelism should be left to those persons who can speak in the same language as the apostle here.
Another important aspect is the motivation the apostle relates for his evangelism: What we've seen and heard, we proclaim to you as well, so that you might have fellowship with us. . . . We write these things to our joy may be complete (vv. 3-4). The purpose of evangelism is to share the fellowship we have with the Holy Trinity, to include others into it as well. As Karl Barth says in CD IV/1, the Christian message is fundamentally "God with us." We have fellowship with God, but as Barth emphasizes, this fellowship belongs to others as well -- they just don't know about it yet. Therefore we bring this message of God's presence among us, and of our fellowship with him, so that the rest may participate in it too.
Of course, one of the critical points here is that evangelism is fundamentally about bringing others into the community of the church. That's why John says, so that you might have fellowship with us. Outside of the context of the church -- and here I am speaking of the whole Christian community, not any particular local congregation -- there is no fellowship with God. That is something which occurs only in Jesus Christ's community. Therefore we do not evangelize so that people remain as they are but believe, nor do we evangelize without integrating the evangelized into a community of believers.
Crucially, too, the apostle relates that the evangelistic endeavor satisfies the longing of the evangelists. They evangelize so that their joy may be complete when others join them in their fellowship. This, I think, is another precondition of evangelism: you must desire the salvation of those whom you are evangelizing. Unless their salvation is a cause of happiness for you, unless your heart burns and aches for them, unless you love them and want to do good to them in gentleness and kindness, don't bother evangelizing anyone.
(There is a provocative question to be asked here: what sort of joy may we have in the next life, if we know that some persons are not enjoying the same fellowship with God which we experience? Wouldn't this detract from our happiness, and from God's as well?)