State estimation (SE) is an important task allowing power networks to monitor accurately the underlying system state, while multi-area SE is becoming increasingly popular as the power grid comprises multiple interconnected 'subgrids.' For nonlinear AC power systems, SE per subgrid amounts to minimizing a nonlinear least-squares cost that is inherently nonconvex, thus giving rise to many local optima. Despite the non-convexity, a recent SE approach based on semidefinite programming (SDP) has been effective in approaching globally optimal performance at the price of higher computational cost. A novel reduced-complexity algorithm is developed in this paper for local control areas to solve the centralized SDP-based SE problem in a distributed fashion. It leverages results on positive semidefinite matrix completion to split a global state matrix constraint into local ones, which further allows for parallel implementation using the alternating-direction method of multipliers (ADMM). With minimal data exchanges among neighboring areas, each control center can efficiently perform local updates that scale with each area's size (number of buses). Numerical simulations using the IEEE 14-bus system demonstrate the asymptotic convergence of local state matrices, and desirable estimation accuracy attainable with a limited number of exchanges.