Why are people able to make death threats with impunity?

Most threats are not crimes. In my state, the legislature has prohibited unconditional threats of immediate harm, coupled with some movement toward carrying ou the threat.

Credible threats are investigated. The vast majority are not credible and resources are not wasted on them.

They ARE illegal. Guys making those threats isolated to their own group of friends is known to be exaggerated no different than the comment "I am so hungry I am starving". It is clearly exaggerated and not literal. People claiming not to be able to see the difference is a fairly common, albeit ridiculous, attack. Serious threats made in public in this regard are illegal and likely to be prosecuted.

If they are reported at the time and investigated and proved

They are investigated. I remember when some men who had threatened President Obama were investigated and found to be in possession of bomb-making materials. They were prosecuted and put in jail. A lot of the "death threats" you hear about are from 15 year old boys who think that sending things on the computer is funny and safe. They pose no real threat.

The problem is catching them. How do you find that 1 person out of a big crowd.

Two reasons.

First, quite often what is being reported is the CLAIM of death threats. The person in questions SAYS they are receiving death threats, but never actualy presents any of them to either the public of the authorities.

Second, its about credibility. The authorities can only prosecute someone if the threat is actually credible. If theres actually a chance that they'll do it, rather than just being mean on the internet. The sort of "death threats" often talked about are social media posts - quite often Tweets - which have absolutely no chance of actually happening.

They're absolutely illegal, and people get in big trouble when they get *caught*. The problem arises when they are made anonymously.