A mother of a friend of mine told me how she poured water over the head of her three grandsons and marked the sign of the cross upon their forehead, saying "I baptize you in the name of the Father and of the Son and of the Holy Spirit." Her daughter married a Jehovah's Witness and promised to raise any future children as Catholics but soon after being married she began considering herself as a Jehovah's Witness and the children never were baptized according to the Trinitarian formula.
From what I understand if a priest is not available the rite can be performed validly and licitly by any person (including a Muslim of an atheist) who has the intention to baptize another and according to the Trinitarian formula. Could someone please provide me with some references from Church documents affirming or correcting this understanding?
His mother, not sure whether she had done the right thing or not, also asked me for some references from sacred Scripture that demonstrate that the laity are allowed to baptize others into the Christian faith. I suggested John the Baptist would be a prime example. Are there any other verses that you can think of?
Thank you for your help in advance.