I agree that death threats should be taken seriously, but we must now reconsider what a death threat is in the age of social media.
Before social media a death threat was typically a phone call or letter, that meant someone had got hold of your contact details and put effort into it.
Someone sending a tweet, whilst obnoxious, saying that "fat burns quicker" is not in the same category of death threat as the above.
If people are sending death threats which contain specific information, such as dealing with time and place then they are serious, or at the very least a form of stalking.
I'm just a bit cautious of everyone saying that they receive death threats these days, particularly as it lessens the impact of people who receive real and detailed death threats.