Descriptively it is private, because you are not sharing it with anyone else - ie, unlike TTN someone can’t sign up to create an account, register devices, contribute gateway traffic, etc.
That depends on the purpose for making the distinction.
For example, in the LoRaWAN hardware-level packet pre-amble it’s not entirely clear if the reserved value is reserved for public LoRaWAN networks or simply for LoRaWAN (vs other forms of LoRa) usage.