There are many methods to podcast but the most popular are:

• Itunes - Load your files to Itunes (an Apple service) and allow the public access then build links back to Joomla on your site.
  
• Feedburner - this is a Google service that makes your podcasts available to all and again you build links on your site (via plugins is best)
  
• Private Podcast Server - setup your own server either within Joomla or externally.

If you want to podcast but not want the podcasts to be too public them the only real solution is a private server.

I experimented with a number of RSS feeders that plugged into Joomla but found most of them difficult for a user and were not Itunes compatible. In the end the best solution (for me) was to setup a separate server and place it in a wrapper within Joomla.

The software I choose was the GNU/GPL Podcast Generator by Alberto Betella . This software is extremely easy to install in a folder on your LAMP server (the same one running your Joomla). The program is fully self contained and writes to its own XML file and so doesn't need your MySQL server. There is builtin the ability to FTP the files to the server in an easy to use interface and before they are uploaded you can add all the headers and categories needed to keep your users informed. To run this software you don't need Joomla at all and an example of this can be found here http://www.podcastgenerator.net/demo/

If you want to use it within Joomla you can create a wrapper but this means you are getting to much styling so you should modify the CSS - as this is relatively simple you can get a good looking result - here is an example - this is a test site and so isn't the best looking, you will find the podcasting example in the main menu.

If you would like the modified CSS you will find it attached. This unzipped folder is placed in /yoursite/yourpodcastfolder/themes/ and selected after logging into the front end and choosing Selecting Themes.