Interfaces should define public procurement , so it does not make sense to have a private member in it.
At least until now. There are probably proposals for C # 8 of the interfaces to allow for implementations and then it would make sense to have private members, although the interface name starts to fade.
At the moment you should simply ignore the private member, if private is implementation detail, then let the class decide on this. When the interface behaves like a trait in C # 8 the private member will be interface detail. It will probably have a protected member, there you can do what you are looking for, but there are controversies about using protected member, even more in an interface, even though it is no longer an interface.
public interface IBar {
string Id { get; }
}
class Foo : IBar {
public string Name { get; private set; }
}