“Each one’s death is his own” seems like an obvious truism, but in fact it’s culturally specific. In the Bible, the dying gathers together his family for final words, blessings, sometimes curses (cf. Genesis 49), and after he dies he is “gathered to his people.” Death is a passage from one community to another.
What is true in the Bible was true for many centuries in the West, and even today families gather for the final days at the hospital bed. What’s different today is that many doubt that there is any community to receive the dead on the other side of the passage. Death is an exit from community, but no longer an entrance.
In any case, the Heideggerian notion that we live toward a death that occurs in isolation is arguably based on modern death practices. Which is ironic, considering Heidegger’s avowed anti-modernism.