If a member of the public does a search, will that display results from the data associated with Unpublished Albums/Images.
The reason this would be useful is that we have a number of images that we do not want to have on public display, but can make available to researchers under particular circumstances. Of course, to do this, a researcher would have to discover that an image exists in his area of interest even if, initially, he cannot view it.

If a member of the public does a search, will that display results from the data associated with Unpublished Albums/Images.

A normal visitor will not get unpublished items. The user would have to be logged in with the appropiate user rights to get unpublished items in search results.

In that case, would it be possible in some future update to have separate "Publish Image with Title & Description" and "Publish Title and Description only" tick boxes? That would enable our member to search for all the images we hold, including the ones to which we do not own the copyright so could only make available under restricted conditions.

Sorry, probably not as an image is either public or it is not. If you want to restrict access like this, require people to login with appropriate rights so they can see them.

There might be perhaps a way to workaround (with the coming 1.5.8) release to hide the actual image from display so it can be search but this would require the images to be published and requires either theme changes or creating a plugin.

Publishing, but hiding the image from display so just the searchable fields are visible to a visitor, sounds a good solution.

Please note this will not be a a ready made feature.