Imagine a sensor network consisting of base stations scattered at random over a large circular region R. Suppose that, in addition, R contains a smaller number of eavesdroppers, also scattered at random. Now suppose that each base station bean monitor a circular area whose radius is given by the distance from b to the nearest eavesdropper e. What is the probability that the entire region R can be monitored securely by the base stations? Extending work in [14], we estimate this probability, and thus provide detailed information on the maximum density of eavesdroppers that can be accommodated, while preserving complete coverage of R. From an engineering perspective, such coverage guarantees a scheme by which mobile stations can roam everywhere in R, and retain communication with a base station. From a mathematical perspective, the results reveal a new and surprising phenomenon, that the obstructions to coverage occur on a wide range of scales.