We study the relation between public and private-coin information complexity. Improving a recent result by Brody et al., we prove that a one-round private-coin protocol with information cost can be simulated by a one-round public-coin protocol with information cost $\le I + \log(I) + O(1)$. This question is connected to the question of compression of interactive protocols and direct sum for communication complexity.
We also give a lower bound. We exhibit a one-round private-coin protocol with information cost $\tilda$ $n/2 - \log(n)$ which cannot be simulated by any public-coin protocol with information cost $n/2 - O(1)$. This example also explains an additive $\log$ factor incurred in the study of communication complexity of correlations by Harsha et al.
