Jigsaw percolation is a nonlocal process that iteratively merges elements of a partition of the vertices in a deterministic puzzle graph according to the connectivity properties of a random collaboration graph. We assume the collaboration graph is an Erdős-Rényi graph with edge probability $p$, and investigate the probability that the puzzle graph is solved, that is, that the process eventually produces the partition $\{V\}$. In some generality, for puzzle graphs with $N$ vertices of degrees about $D$, this probability is close to $1$ or $0$ depending on whether $pD\log N$ is large or small. We give more detailed results for the one dimensional cycle and two dimensional torus puzzle graphs, where in many instances we can prove sharp phase transitions.