  We study the problem faced by a data analyst or platform that wishes to
collect private data from privacy-aware agents. To incentivize participation,
in exchange for this data, the platform provides a service to the agents in the
form of a statistic computed using all agents' submitted data. The agents
decide whether to join the platform (and truthfully reveal their data) or not
participate by considering both the privacy costs of joining and the benefit
they get from obtaining the statistic. The platform must ensure the statistic
is computed differentially privately and chooses a central level of noise to
add to the computation, but can also induce personalized privacy levels (or
costs) by giving different weights to different agents in the computation as a
function of their heterogeneous privacy preferences (which are known to the
platform). We assume the platform aims to optimize the accuracy of the
statistic, and must pick the privacy level of each agent to trade-off between
i) incentivizing more participation and ii) adding less noise to the estimate.
  We provide a semi-closed form characterization of the optimal choice of agent
weights for the platform in two variants of our model. In both of these models,
we identify a common nontrivial structure in the platform's optimal solution:
an instance-specific number of agents with the least stringent privacy
requirements are pooled together and given the same weight, while the weights
of the remaining agents decrease as a function of the strength of their privacy
requirement. We also provide algorithmic results on how to find the optimal
value of the noise parameter used by the platform and of the weights given to
the agents.
