If I banked privately for one child, do I need to do it for additional children?

All the reasons that you banked for the first child are still valid for additional children. 
1. If you want the baby to have the option of using his/her own cells, then you need to bank them. 
2. If you are banking as a form of "biological insurance" for siblings, then the ability to use cord blood from one child for another depends on whether they have matching HLA type.  Two full siblings have a 25% chance of being a perfect match, a 50% chance of being a half match, and a 25% chance of not matching at all.  The more siblings with banked cord blood, the more chance that they cover each other for possible transplants or other therapies for which sibling stem cells are an option.

Reference:
Odds of sibling match are based on haplotype inheritence: that the child will receive 3 HLA types as a group from each parent.