Upper and lower bounds on the expected maximum error for frequency estimation on domains of size B and over n users in different models of DP. The bounds are stated for fixed, positive privacy parameters ε and δ, and $Θ˜/O˜/Ω˜$ asymptotic notation suppresses factors that are polylogarithmic in B and n. The communication per user is in terms of the total number of bits sent. In all upper bounds, the protocol is symmetric with respect to the users, and no public randomness is needed. References are to the first results we are aware of that imply the stated bounds