I don't have to be connected to the internet to view them on my phone. Is it storing the current ones on my device?
The main store of the Photo Stream is on iCloud. Local copies of the latest 1000 photos in your Photo Stream are stored on your iDevices. Photo Stream pictures can also be downloaded into your iPhoto and/or Aperture photo libraries, or onto a designated folder on your Windows PC. Photos are purged from the Photostream cloud store after thirty days, so be certain that photos you want to save are downloaded to your computer, or saved to your iOS device camera roll on a regular basis.
Note that you now have the ability to delete individual photos from Photo Stream Macworld's statement to the contrary was true at the time of publication.
They are stored on your iPhone. You can find them in the Photos app.
It's a smart album filled with photos taken in the last 30 days.
Comment by Jaberg: (thanks!)
The Photo Stream smart album on an iOS device will contain the 1000 latest photos. The main "cloud store"will contain the photos from the last 30 days and those photos will all be viewable/downloadable from a computer.