A decade ago, at a pro-immigration march on the steps of the Capitol building in Little Rock, Ark., community organizer Randi Romo saw a woman carrying a sign that read “no human being is illegal.” She took a photograph and sent it to an activist group, which uploaded it to photo-sharing site Flickr.

Last August, the same image—digitally altered so the sign read “give me more free shit”—appeared on a Facebook page, Secured Borders, which called for the deportation of undocumented immigrants. The image was liked or shared...