Rather than just allow teachers to upload images, we would like to build a feature that does an Internet image search and makes those images available for use in activities. i.e. Search for "apple" and get pictures of apples you can drag into an activity. The source of the image would be acknowledged with a link to the source page, and it would be up to the author of the activity which images to choose.
This feels like copyright infringement, but I'm not so sure. Pinterest, Tumblr and countless others seem to be getting around the liability by offering a mechanism for reporting copyright infringement. What is my actual exposure in the real world? Is the worst I can expect to receive cease and desist notices?
Another part of our system allows users to create their own activities that only they, themselves, would see. Would I need to worry about copyright infringement at all in that case?
Some background first. Under the Digital Millennium Copyright Act (DMCA), service providers are not liable for the copyright infringement of their users provided they follow some procedures outline in the statute. This is usually referred to as the Safe Harbor clause. However, to be classified as a service providers, among other things, they must not be the party providing the content. For example, YouTube may end up hosting infringing content that their users upload, but they do not provide their users with possibly infringing images and music to put in their videos.
Now to your first question. If you are the one providing infringing content for your users to use, you would probably not be covered by the Safe Harbor clause. That means that you could be sued for copyright infringement.
As to your second question, just because it would be hard for the copyright holder to discover that you are infringing their content doesn't make it any less of an infringement. The fact that only one logged in user will see the infringing content may make the odds of getting sued lower, it's probably not a good idea.