I think you don’t understand what an interface is for. Interface is a contract. It specifies that an object behaves in a certain way. If an object implements an interface, it means that you can rely on it that it has all the interface’s methods implemented.
Now, consider what would happen if there was an interface like you’re asking for – public, but with one internal member. What would that mean? An external object could implement only the public methods, but not the internal one. When you would get such an external object, you would be able to call only the public methods, but not the internal one, because the object couldn’t implement it. In other words – the contract would not be fulfilled. Not all of the methods would be implemented.
I think that the solution in this case is to split your interface in two. One interface would be public, and that’s what your external objects would implement. The other interface would be internal and would contain your Save() and other internal methods. Perhaps this second interface could even inherit from the first. Your own internal objects would then implement both interfaces. This way you could even distinguish between external objects (ones that don’t have the internal interface) and internal objects.