  We consider a database $\vec{X} = (X_1,\cdots,X_n)$ containing the data of
$n$ users. The data aggregator wants to publicise the database, but wishes to
sanitise the dataset to hide sensitive data $S_i$ correlated to $X_i$. This
setting is considered in the Privacy Funnel, which uses mutual information as a
leakage metric. The downsides to this approach are that mutual information does
not give worst-case guarantees, and that finding optimal sanitisation protocols
can be computationally prohibitive. We tackle these problems by using
differential privacy metrics, and by considering local protocols which operate
on one entry at a time. We show that under both the Local Differential Privacy
and Local Information Privacy leakage metrics, one can efficiently obtain
optimal protocols; however, Local Information Privacy is both more closely
aligned to the privacy requirements of the Privacy Funnel scenario, and more
efficiently computable. We also consider the scenario where each user has
multiple attributes (i.e. $X_i = (X^1_i,\cdots,X^m_i)$), for which we define
\emph{Side-channel Resistant Local Information Privacy}, and we give efficient
methods to find protocols satisfying this criterion while still offering good
utility. Exploratory experiments confirm the validity of these methods.
