Most congregations will only baptize members of that congregation. There used to be two traveling evangelists in this area who, twice a year, would come around, hold a "revivaL", (They didn't work together, nor were associated), and a baptism, either in the Ohio River, or a local creek, depending upon the weather & water levels. Either of them would baptize anyone who wanted to be baptized. As one of them told me, he didn't question their salvations; that was between each person and Jesus, so he baptized without question.
As for children' being saved or not, the only example we have is David's son by Bathsheba, who died shortly after birth. David, who was a prophet, said he'd join him in heaven some day.
Here you simply fall upon tradition as the function in Baptist congregations you observed.