A friend alerted me she saw my children in a photograph hung in a chain restaurant. Although the picture was taken at a public event, I never gave permission for anyone to sell the image of my children or to hang in such a very public place. Even though it is a public place, there is some expected right to privacy. After all, we don't expect a trip to the grocery store to land us on a billboard. What rights do I have to protect the image of my children? (What if we had been in Witness Protection or hiding from an abusive partner?)
The following is for general information purposes only and should not be taken as legal advice for any particular situation. If you have a specific concern you should consult with an attorney familiar with the relevant issues in the jurisdiction in question. Since the questioner indicated they were located in the U.S., this answer assumes that to be the case.
There are a lot of issues involved here, and any one of them may or may not be the hinge upon which a judge would rule in a case such as this. You need to consult an attorney who specializes in this area and who practices in your jurisdiction and would be familiar with the tendencies of the potential judge(s) that might hear your case were it to go to trial.
That said, there are several considerations that should be made in situations like this.
As to the part of your question that asks,
Depending on what may be deduced from the photo itself as well as how it is displayed, at the very worst the photo of your children only reveals where they were at a specific time in the past and does not necessarily reveal where they presently are. Does the way in which the photo is displayed or captioned even indicate where or when the photo was taken?
I've never been involved with anyone placed in a witness protection program, but I would assume the participants are strongly encouraged to avoid situations, such as attending notable public events, that might reveal their whereabouts. I would also assume the same would be the case for those hiding from an abusive partner.
My answer will be really short, and not based on any law school experience, but only on some conversations about privacy and tort law, etc. and how they apply to photographers, with an uncle who is a lawyer and a cousin in law school, and the fact that I'm a photographer.
As long as the person who took the photo in the public place isn't deriving any profit from the photograph itself by sole virtue of your presence in the photo, or committing libel against you, then your privacy isn't being exploited for others' gain and harm isn't being done to you. So in that case, there would be nothing you could do to them.
Harm can include intangible harm like suffering caused by knowing that an indecent photograph of someone you love is being looked at by others.
"By sole virtue of your presence" means the person alone who is the subject of the photo, rather than the context of the photograph. I.e., I can be paid by the newspaper for my picture of the the winner of the state track meet, and of course that photo consists almost entirely of the person who won. But there is an editorial reason for the picture, i.e., it is a picture "of the winner of the state track meet" rather than "a picture of a person that some other person is hiring me to stalk in public."
Being a celebrity is enough context that people can usually "stalk" celebrities in public legally, but most other people can't be stalked in public without its being construed as harassment.
Sorry, my answer is now getting longer and longer... just look at the second paragraph, and that's all I meant to say.
Photographer here. This isn't a question about privacy, but rather commercial use of your image by a private company on private property which is open to the public. You and your kids have the same rights that a child actor from Modern Family would have. The restaurant hung the picture to enhance their commercial interest. You can't do that without written permission and pay.