Baptism is for cleansing, burial of the old man and identification with Christ - not necessarily for public witness. It is the Scriptural way of expressing a commitment to follow Jesus, and as such it can inspire others to follow Christ - but we should be very wary of decisions to follow Christ purely on the basis of influence of the bold decisions of others. The decision to follow Christ should be based on a recognition of my state as a sinner and the inevitable sentence of death as a result, and by the shocking realization of the love of God in Christ Jesus expressed by taking my place on the cross. This encounter with the love of God is what should evoke the decision to follow Jesus - as our response to HIs love - not any influence, however powerful, by someone else who has had this compelling encounter with the love of God.
Baptism is closely paralelled to marriage. All of the advantages of making marriage public would apply to baptism, however, this does not mean that a private marriage is illigitimate. Since a new believer needs to function immediately in their office as a fully functioning member of the body of Christ, they should only be involved in a public baptism if it can be arranged immediately, and without delay.