  We consider a platform's problem of collecting data from privacy sensitive
users to estimate an underlying parameter of interest. We formulate this
question as a Bayesian-optimal mechanism design problem, in which an individual
can share her (verifiable) data in exchange for a monetary reward or services,
but at the same time has a (private) heterogeneous privacy cost which we
quantify using differential privacy. We consider two popular differential
privacy settings for providing privacy guarantees for the users: central and
local. In both settings, we establish minimax lower bounds for the estimation
error and derive (near) optimal estimators for given heterogeneous privacy loss
levels for users. Building on this characterization, we pose the mechanism
design problem as the optimal selection of an estimator and payments that will
elicit truthful reporting of users' privacy sensitivities. Under a regularity
condition on the distribution of privacy sensitivities we develop efficient
algorithmic mechanisms to solve this problem in both privacy settings. Our
mechanism in the central setting can be implemented in time $\mathcal{O}(n \log
n)$ where $n$ is the number of users and our mechanism in the local setting
admits a Polynomial Time Approximation Scheme (PTAS).
