Given an infinite connected regular graph G=(V,E), place at each vertex Pois(λ) walkers performing independent lazy simple random walks on G simultaneously. When two walkers visit the same vertex at the same time they are declared to be acquainted. We show that when G is vertex-transitive and amenable, for all λ>0 a.s. any pair of walkers will eventually have a path of acquaintances between them. In contrast, we show that when G is non-amenable (not necessarily transitive) there is always a phase transition at some λc(G)>0. We give general bounds on λc(G) and study the case that G is the d-regular tree in more details. Finally, we show that in the non-amenable setup, for every λ there exists a finite time tλ(G) such that a.s. there exists an infinite set of walkers having a path of acquaintances between them by time tλ(G).EPSRC grant EP/L018896/1