Existing network switches implement scheduling disciplines such as FIFO or deficit round robin that provide good utilization or fairness across flows, but do so at the expense of leaking a variety of information via timing side channels. To address this privacy breach, we propose a new scheduling mechanism for switches called indifferent-first scheduling (IFS). A salient aspect of IFS is that it provides privacy (a notion of strong isolation) to clients that opt-in, while preserving the (good) performance and utilization of FIFO or round robin for clients that are satisfied with the status quo. Such a hybrid scheduling mechanism addresses the main drawback of prior proposals such as time-division multiple access (TDMA) that provide strong isolation at the cost of low utilization and increased packet latency for all clients. We identify limitations of modern programmable switches which inhibit an implementation of IFS without compromising its privacy guarantees, and show that a version of IFS with full security can be implemented at line rate in the recently proposed push-in-first-out (PIFO) queuing architecture.