This paper investigates physical layer security for a large-scale WSN with random multiple access, where each fusion center in the network randomly schedules a number of sensors to upload their sensed data subject to the overhearing of randomly distributed eavesdroppers. We propose an uncoordinated random jamming scheme in which those unscheduled sensors send jamming signals with a certain probability to defeat the eavesdroppers. With the aid of stochastic geometry theory and order statistics, we derive analytical expressions for the connection outage probability and secrecy outage probability to characterize transmission reliability and secrecy, respectively. Based on the obtained analytical results, we formulate an optimization problem for maximizing the sum secrecy throughput subject to both reliability and secrecy constraints, considering a joint design of the wiretap code rates for each scheduled sensor and the jamming probability for the unscheduled sensors. We provide both optimal and low-complexity sub-optimal algorithms to tackle the above problem, and further reveal various properties on the optimal parameters which are useful to guide practical designs. In particular, we demonstrate that the proposed random jamming scheme is beneficial for improving the sum secrecy throughput, and the optimal jamming probability is the result of trade-off between secrecy and throughput. We also show that the throughput performance of the sub-optimal scheme approaches that of the optimal one when facing a stringent reliability constraint or a loose secrecy constraint.