  Graph spanners are sparse subgraphs which approximately preserve all pairwise
shortest-path distances in an input graph. The notion of approximation can be
additive, multiplicative, or both, and many variants of this problem have been
extensively studied. We study the problem of computing a graph spanner when the
edges of the input graph are distributed across two or more sites in an
arbitrary, possibly worst-case partition, and the goal is for the sites to
minimize the communication used to output a spanner. We assume the
message-passing model of communication, for which there is a point-to-point
link between all pairs of sites as well as a coordinator who is responsible for
producing the output. We stress that the subset of edges that each site has is
not related to the network topology, which is fixed to be point-to-point. While
this model has been extensively studied for related problems such as graph
connectivity, it has not been systematically studied for graph spanners. We
present the first tradeoffs for total communication versus the quality of the
spanners computed, for two or more sites, as well as for additive and
multiplicative notions of distortion. We show separations in the communication
complexity when edges are allowed to occur on multiple sites, versus when each
edge occurs on at most one site. We obtain nearly tight bounds (up to polylog
factors) for the communication of additive $2$-spanners in both the with and
without duplication models, multiplicative $(2k-1)$-spanners in the with
duplication model, and multiplicative $3$ and $5$-spanners in the without
duplication model. Our lower bound for multiplicative $3$-spanners employs
biregular bipartite graphs rather than the usual Erd\H{o}s girth conjecture
graphs and may be of wider interest.
