No, they should not be public unless there is a clear need for public protection.
Far too many people are censorious and judgemental and the universal, in its full sense, change of attitude that would mean that everyone would smile benignly on youthful follies and help the reformed criminal make a new life will never happen.
Does a young man, who was involved in mindless vandalism at 18, want everyone knowing about and judging him when he is 50? The woman caught shop-lifting at 40, coping with grandchildren taunted at school because grandma is a thief?
There is nothing dishonest about this. Most of us sometime in our lives has done something non-criminal that we are deeply ashamed of. We learn from our mistake and put it behind us. Would you want everyone to know, that you were part of a gang of bullies at school, or stole money from your grandmother's purse?
Those who need to know will be told of criminal convictions when necessary. I am quite happy with that.