  In distributed interactive proofs, the nodes of a graph G interact with a
powerful but untrustable prover who tries to convince them, in a small number
of rounds and through short messages, that G satisfies some property. This
series of interactions is followed by a phase of distributed verification,
which may be either deterministic or randomized, where nodes exchange messages
with their neighbors.
  The nature of this last verification round defines the two types of
interactive protocols. We say that the protocol is of Arthur-Merlin type if the
verification round is deterministic. We say that the protocol is of
Merlin-Arthur type if, in the verification round, the nodes are allowed to use
a fresh set of random bits.
  In the original model introduced by Kol, Oshman, and Saxena [PODC 2018], the
randomness was private in the sense that each node had only access to an
individual source of random coins. Crescenzi, Fraigniaud, and Paz [DISC 2019]
initiated the study of the impact of shared randomness (the situation where the
coin tosses are visible to all nodes) in the distributed interactive model.
  In this work, we continue that research line by showing that the impact of
the two forms of randomness is very different depending on whether we are
considering Arthur-Merlin protocols or Merlin-Arthur protocols. While private
randomness gives more power to the first type of protocols, shared randomness
provides more power to the second. Our results also connect shared randomness
in distributed interactive proofs with distributed verification, and new lower
bounds are obtained.
