How much space do "Shared Streams" take up on an iOS device?

Solution 1:

• When others share with me, are the photos copied to my hard drive? Or do they purely reside in the cloud?

Photos shared with you (or photos you share) get copied to your (or others') devices through iCloud. They reside on iCloud for just 30 days to give adequate time for all devices to sync them from the cloud. See iCloud: My Photo Stream FAQ

• If I add the same photo to two different streams, will it duplicate the amount of space used on my phone? Or does Apple intelligently make a reference link instead of a copy?

It won't add another copy on your phone across your photo streams, but there will be a photo stream specific copy that's separate from the camera roll (and any other albums the photo may be part of).

The quality of the photos shared will also vary across devices. See iCloud Photo Stream and Image Quality

There are three groups you see under Settings > General > Usage > Photos & Camera:

1. Camera Roll - this is for photos you have taken on the device as well as photos you have saved to the Camera Roll from (others') photo streams you have subscribed to. If you have Photo Stream enabled, any photo you take with your camera will be added to the Camera Roll and Photo Stream (with Photo Stream being restricted to the recent 1000 photos on iOS devices; there are no restrictions on photo streams on computers).
2. Photo Library - this is for photos that have been synced from iTunes (the source of those usually being iPhoto or Aperture on OS X).
3. Photo Stream - this is for photos from your and others' photo streams. If you delete a photo from your photo stream, it will get deleted from all other devices that sync to the same iCloud account.

Solution 2:

As of my experience the whole Photostream feature needs about 1GB of space on your iOS device.

When I reinstalled my iPhone recently and haven't taken any photo yet - iTunes showed up a usage of ≈1.1GB for photos, just because I had the Photostream activated.

The interesting thing is, that I had many photos in my Photostream, that were made with my DSLR and synced into the Photostream via iPad. All of these ≈600 Photos would have used a lot more that these 1.1GB, so I guess you get a smaller version of your photos for your iPhone. However you get the pictures in full-size in iPhoto on a Mac.