The Maximum Weight Matching (MWM) problem is solvable in polynomial time through Edmond's famous ``blossom" algorithm. The MWM problem can also be stated as the solution to a particularly chosen sequence of Linear Programs (LPs). However, known implementations of the ``blossom" algorithm are not distributed. We consider a Graphical Model with factors limiting the number of edges incident to a vertex, as well as factors enforcing ``blossom" inequalities and construct distributed algorithm of the popular Belief Propagation type which converges to the MWM solution iff the corresponding LP is tight and unique. This motivates the development of a sequential, distributed algorithm for MWM. We experimentally compare the proposed algorithm to state-of-the-art ``blossom" algorithms.