No, it is not true at all. Have you not heard all the news of "revenge porn" sites being destroyed by new laws and law suits? The jist of it is that the subject of an image, whether or not they gave consent of the image being taken, has the right to deny it's distribution. You bet the neighbor would win in court, people are constantly winning these cases.
To the OP; go straight to the management company. You would be surprised at how a proper business will stand up straight and take action when bad publicity rears its ugly head and looks them in the eye. All non monopolized businesses are ultimately at the mercy of reputation, and threatening a lawsuit or action through local real estate agents and listers will get their attention (particularly if you had neighbors with similar stories, the more you can pile at their door the better in this case). The worst thing you can do in this situation is stand by quietly and hope things get better.
By the by if you keep you animals in your our their own room, I think you would be allowed to put a lock on that door. May need to double check your lease, but I don't think they usually cover anything besides the front door. Could be wrong on this, but worth checking in any case.