When Jesus exorcises the demoniac near the tombs, he doesn't permit him to become one of his followers, but he says: Return to your home, and declare how much God has done for you (Luke 8.39). He was from a town of Gentiles; he goes and begins to prepare his hometown for the gospel of Jesus Christ, which will eventually spread to the Gentile lands and peoples after the resurrection.
This is an example of how testimony about the works of God function in evangelism. This sort of thing is present in many of the psalms, as well. Consider for example:
Many will see and fear,
and put their trust in the LORD. . . .
I have told the glad news of deliverance
in the great congregation;
see, I have not restrained my lips,
as you know, O LORD.
I have not hidden your saving help within my heart,
I have spoken of your faithfulness and your salvation;
I have not concealed your steadfast love and your faithfulness
from the great congregation (Ps 40.3, 9-10).
The psalmist experiences God's deliverance in some impressive way, but he doesn't keep the event all to himself. Rather he speaks out about it, and tells the whole congregation. He says, moreover, that this testimony has an evangelistic function. It will move many to see and fear, and put their trust in the LORD (v. 3).
There are numerous other psalms that speak about people turning to God after having witnessed the great things he had done. This is most notable in Ps 22, which speaks of the results of the redemption of the suffering servant of God:
All the ends of the earth shall remember
and turn to the Lord;
and all the families of the nations
shall worship before him. . . .
Posterity will serve him;
future generations will be told about the Lord,
and proclaim his deliverance to a people yet unborn,
saying that he has done it (vv. 27, 30-1).
I think the psalmist is speaking about the true suffering servant, Christ. More precisely, I believe his words find prophetic fulfillment in the resurrection of Christ, which will be followed by the conversion of the nations. Every knee will bow down to him (Ps 22.29; Phil 2.9-11) and worship Christ! This is because they will learn of what God did for him, how he raised him from the dead. They will recognize what they did wrong, and they will acknowledge that he is the true Son of God.
This motif, then, of evangelism by personal testimony is prevalent in the scriptures. Of course, we should always focus our evangelism on the testimony of the apostles to the resurrection of Christ, since that is what saves us, above everything else. Still there is place for the mention of our own experiences of God's deliverance in the midst of it all, as well. We have ample biblical precedent for such a practice, both in evangelism and in the context of the church meeting.
One of the things I like about the Romanian Pentecostal churches is the way they do intercessory prayer. Every prayer meeting, there is one specific prayer dedicated to intercession on behalf of those suffering or troubled in some way, whether present in the congregation or not. The whole assembly is given the opportunity to mention reasons for prayer with a loud voice, and after they have all been announced, everyone prays together for those things. Here, too, there is opportunity to bring thanksgiving before God and the assembly for things which have taken place in the lives of the members of the church. I find that messages of thanksgiving always motivate the others to pray, because they are newly encouraged and reminded that God is still active and working in the world.
Telling others about what the Lord has done for you is a way of letting them know that God is still out there; he still cares for us. If he answered my prayer, he can answer yours. I prayed for a long time that I would find a place of work which was suitable to my talents and my training. Finally, after more than a year of praying, I have found a job at Grand Canyon University. Furthermore it is a job which offers the opportunity of promotion in the future. For this, I am endlessly thankful to God, because this job is an undeniable gift. I enjoy what I do immensely; there could not have been a more perfect place or time for me to get this job. In the same way, I encourage everyone else to pray to God and to trust that he will provide for you.