What does Public and Private status mean for supplier users?

Public means that the supplier organization user that exists in your realm has registered with Ariba Network and is connected to a user on Ariba Network.

Private means that the supplier organization user that exists in your realm is not connected to a user on Ariba Network. Users remain with a status of Private until they either register with Ariba Network for Suppliers or your site synchronizes their existing registration.

See KB0405574 - How do I know if a supplier is Private or Public?

For more questions about the supplier registration, check out KB0396574 - Common Sourcing Supplier Registration Scenarios

SAP Business Network for Procurement & Supply Chain > Administration (Buyer & Supplier Account Administration)