Hypothetically, if a good or service were to be sold on the black market, would that count as a private good that is in a sense not rivalrous due to it being illegal, but also rivalrous in the context of buying it on the black market?
The qualification of private good has to do with whether or not the item is rivalrous (one's consumption decreases the value of another's consumption) and excludable (whether or not you can prevent people from benefitting from your good if you provide it for one person). Private goods are rivalrous, if I consume it, there is less for you. They are also excludable: if I provide one for person A, I can ensure that only person A or anyone person A shares it with consume it. When we say something is provided on the black market, we are just saying that the buying/selling of this item is happening outside of legal parameters.