There are some training techniques that rely on negative examples to improve classification, though I'm not sure they exist for images
Well, this can somewhat be accomplished by classifying exactly what an image isn't , but as you can imagine, that is (in the general case) a combinatoric nightmare.
You can just do binary classification. That makes the decision boundary a bit more complicated, but if you're using neural networks, that is solved by adding another layer at the end.
The "problem" here is probably not the lack of negative training data, but just that they want as few false negatives as possible, so they use a very sensitive model. That naturally increases the number of false positives. It's still a win if you only have to look at some desert pictures instead of literally every picture on someone's computer.