Argthtjtr

Something like the reciprocal of 'trustworthy'.

'You need to understand how [requiring a high degree of trust] it is
to get to know the code to the safe. John has only told three people
that code ever because of the power of what's behind the vault's
doors'

Almost 'sensitive' but with 'power' and 'responsibility'

'The position of babysitter is not a trivial one, as it is a
[requiring a high degree of trust] relationship.'

Almost 'sacred' but not quite--it needs to convey the level of responsibility and power that a babysitter has over a child.

'A pediatrician must not only be knowledgeable but kind as he holds
the wellbeing of a child in his hands, his position is [requiring a
high degree of trust]'

1. 'You need to understand the certitude one requires to get the
   code to the safe. John has only told three people that code ever
   because of the power of what's behind the vault's doors'




2. 'The position of babysitter is not a trivial one, as it requires a
   high degree of certitude.'




3. 'A pediatrician must not only be knowledgeable but kind, as he holds
   the well being of a child in his hands. His position is one of
   certitude.'

certitude TFD

n.
1. The state of being certain or convinced of something; complete assurance; confidence; certainty; total certainty or greater certainty than circumstances warrant.

The word you need is "trustworthy", but you will need to modify your sentences slightly to include that e.g. "The position of babysitter is not a trivial one - it requires trustworthiness".

Or "You need to be trustworthy to be given the keys to the safe".

The higher the cost or consequence of eventual betrayal the higher the degree of trust is required.You don't have to trust someone that has no way of betraying the trust by hurting you or causing you loss.
Example: You would lend someone you knew briefly 5 dollars but you wouldn't lend him 100 dollars.

Higher degree of trust we usually assign to people with good REPUTATION.
Not only is this word used in context of moral/unmoral lifestyle but it depends of the role of a person in society. Reputable science laboratory, school, manufacturer may count on higher degree of our trust.

Intimate fits the bill for most of these sentences.

2 : of a very personal or private nature

https://www.merriam-webster.com/dictionary/intimate