Adaptive finite element methods (AFEM) have been successfully used in numerical solutions of partial differential equations (PDEs) yielding optimal convergence rates with respect to degrees of freedom. However, due to the nature of AFEM, each refinement step requires a new set of computations leading to cumulated work. One then strives to achieve optimality with respect to overall cost, i.e. total elapsed time. The core question becomes to design contractive algebraic iterative solvers used within AFEM. In the context of symmetric linear elliptic second order PDEs, we propose a local adaptive multigrid solver, where the linear system stems from a finite element discretization with polynomial degree p and bisection-generated meshes with local size h. The proposed solver contracts the algebraic error hp-robustly and comes with a built-in a posteriori error estimator. This estimator provides a two-sided bound of the algebraic error. More precisely, proving the hp-robust contraction of the solver is in fact equivalent to showing that the built-in estimator provides an hp-robust upper bound on the algebraic error. Moreover, the error-equivalent estimator and its localized decomposition leads to the development of an extension of the solver with an adaptive number of additional local smoothing steps assuring further error contraction. Presented numerical results highlight the performance of the solver, its adaptive version, and optimality of the full algorithm.