What you're looking for is at the intersection of:
1. "Save As..." for a website, where some of the files (HTML, images) on a specific page are downloaded, but other media (HTML5, Flash, JS, ASP) does not get downloaded.
2. website crawler, which downloads everything from the site (not sure about HTML5, Flash, JS, or ASP)
I'm not exactly sure how important is to save the content as is, rather to allow users to see what I see when I want to share with them something. Sort of advanced page/screen-shot taker with easy sharing of such shots.
Which part of the content on the page are other users prevented from accessing? This is the content you'll essentially be rehosting / hosting locally.
Another way to do this would be having a service to rip and rehost content, somewhat similar to a FLV-ripping site (ie. download from Youtube). This kind of site would likely violate copyrights on the rehosted content. You could provide your user/pass to another site, then this service could use your creds to crawl a restricted site and constantly update itself -- but how many people would trust a random crawling site?
1. "Save As..." for a website, where some of the files (HTML, images) on a specific page are downloaded, but other media (HTML5, Flash, JS, ASP) does not get downloaded.
2. website crawler, which downloads everything from the site (not sure about HTML5, Flash, JS, or ASP)