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]'