We study on the initial-boundary value problem for the coupled degenerate hyperbolic system with dissipation~: \begin{align*} \begin{cases} & \displaystyle \rho \frac{\partial^2 u}{\partial t^2} -\left(\int_\Omega |\nabla u(x,t)|^2\,dx + \int_\Omega |\nabla v(x,t)|^2 \,dx \right) \Delta u+ \delta \frac{\partial u}{\partial t} =0 \,, \\[1em] & \displaystyle \rho \frac{\partial^2 v}{\partial t^2} -\left(\int_\Omega |\nabla u(x,t)|^2\,dx + \int_\Omega |\nabla v(x,t)|^2 \,dx \right) \Delta v+ \delta \frac{\partial v}{\partial t} =0 \\ \end{cases} \end{align*} with $\rho>0$ and $\delta>0$ and a homogeneous Dirichlet boundary condition. When either the coefficient $\rho$ or the initial data are appropriately smaller than the coefficient $\delta$, we show the global-in-time solvability for the system and the optimal decay rate for the $H^2$-norm for the solutions. Moreover, we derive the sharp decay estimates of their derivatives.