If I want to set my property in a Class private, so it should be only possible to use and set this property in this class, what is the better way? This
public string Name { private get; private set }
or
private string Name { get; set }
hmmm and there is also
private string Name { private get; private set }
Have you tried compiling your examples? Only the middle one will translate.
If you want to specify extra accessibility level keyword, you can only do it on one of the accessors (getter/setter), and that level of the accessor must be more restrictive than the accessibility of the entire property.
Here you see the rules: Restricting Accessor Accessibility