How to securely authenticate content embeded in another site?
1- build the content structure well in your first site and create a view from admin/structure/views/add that return the data needed to the second site which acts like a client consuming your first site content.

2- Download and enable Services module and enable the submodule Rest Server and submodule OAuth Authentication now you can create secured resources to be consumed by any other sites.

4- create an end point from admin/structure/services/add and choose authentication type as your case "OAuth Authentication" will fit as you need to authenticate the client "the second site consumes your data".

3- Download and enable Services views module that will help you expose any view as a resource to be consumed and managed via services and 

4- From the view you have created before click the button Add and choose services, now the view will appear as a regular resource in your site under admin/structure/services/list click edit resources to see it named with your view name enable it and choose the required authentication and authorization.

Now any authenticated client client site or App can consume that resource from http://YOUR_SITE_DOMAIN/END_POINT/VIEWS_SERVICES_URL.

The last and the most important part in your requirements is to configure the authentication method and test it will I recommend below helpful tools and articles.

Chrome rest client extension to test the webservice.

Very helpful article about services module

Another helpful article