The family of Terri Schiavo had a very difficult decision to make. The news media could not adequately portray all of the factors that went into their decision making, which meant that on top of the tragedy of Schiavo's medical situation, they had perfect strangers passing judgment on them. Families with trouble need privacy so they can make the best decisions possible under the circumstances.
Yes they should have had more privacy. When you are dealing with death of a loved one it takes a toll on the whole family. You have raw emotions and you don't really want to be bothered by people during this time but only family and close friends and maybe a pastor.
The family of Terri Schiavo should have gotten more privacy just because the family's medical decisions should have been private. The court case was very public, unfortunately. The media overstepped its bounds in regards with regards to the Schiavo case. Yes, it was a defining moment in medical cases. However, the public didn't have a right to know that much.
The family of this young woman was raising an important question by disputing whether she should or should not be taken off life support. So I do not know if they asked for more privacy, but they very well may have been happy that this kind of attention was brought to the issue.
In the Terri Schiavo case, Terri Schiavo was in a vegetative state. Her parents wanted her kept on life support, while her husband wanted it removed. Her parents wanted the publicity, because they felt that they were not being given the right to keep their daughter alive. They provided video, for example, to show that they felt she was not completely brain dead. In this case, it wasn't merely the media that wanted it made public, but also her family. It has actually brought some good debate to the public and helped increase awareness of end-of-life issues.