Praise in public, chastise in private
The phrase "praise in public, chastise in private" refers to the philosophy in child raising, that being willing to let anyone know your praise for the child's accomplishments, but avoiding public humiliation by keeping mistakes and any punishments between guardian and ward.
If "chastise" means corporal punishment such as a spanking, the principle demands that the same should take place in privacy, with no other people watching and listening. So if an offense is committed in the presence of others (e.g. in a full classroom or in any public place such as a shopping mall), punishment will be not on the spot but will be given maybe later at home, or perhaps in the semi-privacy of a public restroom. Some will hold differing standards of what counts private differing between siblings and other peers. Where some would a witnessing sibling still counts as private being in the family and other would disagree.
The same principle also often finds application in public relationships between adults, such as in workplace situations, only that "chastise" will refer to things like criticism or firing by an employer.