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Here’s a simple hack that will help you subscribe and follow any private feeds inside Google Reader provided you know the username and password that is required to view that feed in the browser.

Google Reader and Private RSS Feeds

Google Reader doesn’t allow feeds that require authentication but you can easily view such feeds inside services like Yahoo Pipes, Netvibes or even FeedBurner by adding the username and the password to the feed URL itself — see example:

http://user:password@abc.com/rss-feed

It works like this. You can pass a private RSS feed to FeedBurner or Yahoo! Pipes using the above syntax, they’ll will automatically convert your feed into a public feed and you can then subscribe to the new feed inside Google Reader without requiring any username or password.
For instance, this RSS feed from Twitter requires authentication so you won’t be able to view the contents of the feed unless you know the account credentials. You can however subscribe to the same feed in Google Reader using either FeedBurner or Yahoo! Pipes.
It’s an easy workaround but you may not feel very comfortable as you have to add the password in plain text to the feed URL itself. Second, search engines may index your feed, you should definitely not use this with your Gmail Inbox or your private Flickr photos. Use FeedDemon instead.
That said, if there are feeds that are protected by password but don’t really contain any private data (say an invitation-only blog), here’s how you can subscribe to such feeds inside Google Reader.

1. Subscribe to Private Feeds via FeedBurner

The following video screencast shows how you can use FeedBurner to subscribe to private feeds inside Google Reader.
Once you have created the feed, go to the “Publicize” tab in FeedBurner and turn on the NoIndex option so that your private feed does not get indexed in search engines.

2. Make Private Feeds Public with Yahoo! Pipes

Here’s a second screencast that illustrates how you can make a password-protected feed public with the help of Yahoo! Pipes.
When your “pipe” is complete, don’t publish it so it won’t appear on your Yahoo Pipes profile but you’ll still be able to subscribe to the converted RSS feed in any newsreader.
The advantage with Yahoo! Pipes is that is provides direct JSON output so developers can do more interesting things with the feed data.