References to ‘private’ baptisms are to brief ceremonies, often performed by someone other than the minister of the parish and usually at the place of birth, for a child who was not expected to survive.
A church baptism ceremony included the child’s being ‘received into the church’, so this part was not included in a private baptism. If the child survived, the formalities were completed by a church ceremony – sometimes recorded in the parish register without explanation – to receive it into the church.
This can explain why some children appear to have been baptised twice.