When can several people simulate a private random oracle for someone else?

Many models of secure multiparty computation supply each participant with a private random oracle [Yao82,Sha79,GMW87,Csi08]. If one participant does not have access to a private random oracle, perhaps the other participants may work together to simulate one. We prove a complete set of restrictions for this simulation. If two or more participants have suitably strong random oracles that may sample infinite probability spaces, then they may simulate full-strength random oracles for everyone else, subject to mild conditions on the probability spaces. If their random oracles are restricted to finite probability spaces, two participants may simulate, at best, random oracles which take probabilities in the rationals; three or more participants may simulate, at best, those which take probabilities in the algebraics. These results improve on those laid out by Yao in his seminal paper [Yao82].