Please note that this behavior is proper and by design. Web parts and their setting are binded to the page template and are shared on all pages, using the same template. This behavior concerns web parts itselves, and its properties (in general everything what you can see/change on Design tab). It is not concern to Page mode, so for example content of some web parts (editableText, editableImage, ...) is page independent and can be different even on shared page template. You can solve it if you create ad-hoc template (Properties -> Template). Then you will get independent, but the same templates, and you can change iis content for particular page, with no influence to other pages. For more details please see this knowledge base article.