Lower bounds on alternative models of multiparty communication complexity
The "number in hand" model is the one I was thinking of, and there is quite a bit of literature about it. In particular I found this paper of Jeff Phillips, Elad Verbin, and Qin Zhang from SODA 2012 [1]. In particular they prove lower bounds on the problem I was interested in using, the undirected graph connectivity problem, of $\Omega(nk / \log^2(k))$. Here $k$ is the number of edges provided to each player and $n$ is the number of vertices in the graph.

[1] http://arxiv.org/abs/1107.2559