This paper addresses a task allocation problem for a large-scale robotic swarm, namely swarm distribution guidance problem. Unlike most of the existing frameworks handling this problem, the proposed framework suggests utilising local information available to generate its time-varying stochastic policies. As each agent requires only local consistency on information with neighbouring agents, rather than the global consistency, the proposed framework offers various advantages, e.g., a shorter timescale for using new information and potential to incorporate an asynchronous decision-making process. We perform theoretical analysis on the properties of the proposed framework. From the analysis, it is proved that the framework can guarantee the convergence to the desired density distribution even using local information while maintaining advantages of global-information-based approaches. The design requirements for these advantages are explicitly listed in this paper. This paper also provides specific examples of how to implement the framework developed. The results of numerical experiments confirm the effectiveness and comparability of the proposed framework, compared with the global-information-based framework.