In this report, we study the multiparty communication complexity problem of the multiparty equality function (MEQ): EQ(x_1,...,x_n) = 1 if x_1=...=x_n, and 0 otherwise. The input vector (x_1,...,x_n) is distributed among n>=2 nodes, with x_i known to node i, where x_i is chosen from the set {1,...,M}, for some integer M>0. Instead of the "number on the forehand" model, we consider a point-to-point communication model (similar to the message passing model), which we believe is more realistic in networking settings. We assume a synchronous fully connected network of n nodes, the node IDs (identifiers) are common knowledge. We assume that all point-to-point communication channels/links are private such that when a node transmits, only the designated recipient can receive the message. The identity of the sender is known to the recipient. We demonstrate that traditional techniques generalized from two-party communication complexity problem are not sufficient to obtain tight bounds under the point-to-point communication model. We then introduce techniques which significantly reduce the space of protocols to study. These techniques are used to study some instances of the MEQ problem.