Numerical investigation of the underexpanded sonic coaxial jets is carried out using large eddy simulation for three typical inner nozzle lip-thicknesses. Various fundamental mechanisms dictating the flow phenomena including shock structure, shear layer evolution and sound production are investigated. It is found that the inner nozzle lip induces a recirculation zone between inner and outer jets, which significantly influences the behaviors of shock structures and shear layers. The sound properties of the coaxial jets are further analyzed in detail. As the inner lip-thickness increases, the helical screech mode switches to an axisymmetric one and high-frequency screech also occurs with an oscillation frequency of recirculation zone. Based on the temporal Fourier transform and correlation analysis, the primary sources of low- and high-frequency screeches are associated with the downstream shock cells in the jet column and the secondary shock structures in the outer annular jet, respectively. The proper orthogonal decomposition analysis reveals that the dominant structures constructed by the most energetic modes shift from the downstream shock cells region to the upstream secondary shock region as the lip-thickness increases. The results obtained in this study provide physical insight into the understanding of the mechanisms relevant to the coherent structures and sound properties in sonic coaxial jets.