We address the problem of distributed estimation of eigenvalues in power system models using synchronized phasor measurements. The power system is partitioned into a set of non-overlapping areas, each of which is equipped with a local estimator. Online measurements of bus voltage and current phasors from a limited number of buses in each area are used for identifying the characteristic polynomial of the system in a distributed fashion by exchanging information between these local estimators and a central estimator. We develop a new variant of distributed least squares for carrying out this estimation, and show that the algorithm converges with exponential speed. The identified characteristic polynomial is finally used for estimating the eigenvalues. Both synchronous and asynchronous versions of the algorithm are proposed. The algorithm can be implemented without any matrix inversion with minor modification. Results are validated using the IEEE 68-bus power system model with five areas.