Stochastic computing (SC) performs arithmetic on randomized bit-streams called stochastic numbers (SNs) using standard logic circuits. SC has many appealing features such as error tolerance, low power, and low area cost. However, it suffers from severe accuracy loss due to correlation or insufficient randomness. SNs can be decorrelated by regenerating them from independent random sources. This is the preferred decorrelation method mentioned in the literature, but it often entails huge area and delay overhead. An attractive alternative is isolation-based decorrelation, which is the focus of this research. Isolation works by inserting delays (isolators) into a stochastic circuit to eliminate undesirable interactions among its SNs. Surprisingly, although it has far lower cost than regeneration, isolation has not been studied systematically before, hindering its practical use. The paper first examines the basic characteristics of SC isolation. We show that unless carefully used, it can result in excessive isolator numbers or unexpectedly corrupt a circuit's function. We therefore formally characterize the behavior of an isolation-decorrelated circuit, and derive conditions for correct deployment of isolators. We then describe the first isolator placement algorithm designed to minimize the number of isolators. Finally, we present supporting data obtained from simulation experiments on representative circuits.