What's a good word to describe the situation when sensitive information is given to a trusted proxy (such as a lawyer), to be released only when a certain event occurs, such as a person's untimely death?
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TV Tropes (warning: not productivity safe) has an entry for Dead Man Writing, and some of those examples cover what you're talking about.
Edit: Answered before the question was clarified, and addressing the information so entrusted, rather than the process.
If the context is already established "insurance" is not a bad choice. But you can't use it without setting the stage because it has too many alternate meanings.
On a snarky note, there is "last will and testament", but again it won't work if you haven't set it up.
This would actually be the "testament" part of last will and testament. It comes, through three other forms, from the Latin for witness, which seems to dovetail pretty perfectly into the heart of the situation you're trying to describe. The act of putting that information in the hands of a chosen proxy would be described as giving testament.
The second google definition looks perfect.