In the decarbonization in the electricity sector, the emergence of prosumers in the medium/low voltage power distribution systems poses significant challenges for distribution system operators (DSOs), as bidirectional power exchange and carbon trading occur more frequently. This paper focuses the coordinated electricity and carbon management in a bi-level frame-work for DSOs and local distributed resources aggregators (DRAs). At the upper level, the optimal power flow (OPF) and carbon emission flow model (CEF) are solved to derive the locational marginal price (LMP) and nodal carbon intensity (NCI) for DRAs. At the lower level, an exponential-based pricing model is designed to enable peer-to-peer (P2P) carbon emission permit (CEP) trading among DRAs, which involves CEP demand-supply situation and emission assessment pressure. Chance constraints are formulated to address uncertainties in DRAs. Numerical results confirm the effectiveness of the proposed solution in coordinated electricity and carbon operation of multi-stakeholders in power distribution systems.