No, the photos stay in the shared albums. They download to your device, but only for viewing. If the owner of the album deletes the photos, you will lose access to them.
If there are photos in a shared album that you really like, you should save them to your own library. Once there, you have them for good regardless of what happens to the shared album. Note that the saved photo will show up in its appropriate location date-wise based on when it was taken, so you may not see at the bottom where you'd expect. Go into "All Photos" album to find it at the bottom.