You control how miners work on your own private network.
Generally speaking, private networks are implemented specifically to isolate your environment.
This could be due to wanting to secure your data from the public network or simply because you want to have an environment for dev/QA.
Therefore, you wouldn't want any unknown miners attached to your private network. One use case that may cause you to want expose your network to other entities would be if you were creating a consortium that others could join. Something like sellers joining an Amazon private network in order to handle fulfillment. Even in this case, the miners on the network would be setup in a way similar to traditional private networks today where access is strictly controlled by the owner of the network (with firewalls, etc).
If you're interested in setting up your own private network, the Geth documentation has a pretty good write up.