While they should be, people are not being held responsible for any number of foolish, reckless actions that they are doing today. Especially rampant is the anonymous bullying and hacking done today. Celebrity deaths are just another notch on the board of an anonymous user who likes the feeling of power.
Police have never arrested anyone for starting rumors of celebrities death, maybe the reason is because such rumors have become rampant making it hard to be tamed. But that said, they ought to arrest the rumor-mongers because such reports have psychological effect on the celebrities and their families, lets not forget that celebrities are people too and have feelings just like other people.
There has never been a case successfully prosecuted where a person was charged with slander for starting a rumor about a celebrity. There have been some court cases involving slander, but those were about direct comments made by the individual in the media. People routinely get by with starting false rumors about celebrities.
No, people are not held responsible for starting rumors of a celebrity's death. People who are close to the celebrity, however, would know whether or not the rumors are true. The people who tend to believe the rumors are fans who are not personally connected to the celebrity. This sin't that bad.