Let \(G\) be a compact abelian group with dual group \(\Gamma\) and normalized Haar measure \(\mu\), let \(E = \{\gamma_n\} \subset \Gamma\) be a Sidon set with the Sidon constant \(S(E)\). The authors prove the existence of positive \(c = c(S(E))\) such that the inequalities \[ c^{-1} P [|\sum^N_{n = 1} a_n r_n |\geq c \alpha] \leq \mu [|\sum^N_{n = 1} a_n \gamma_n|\geq \alpha ] \leq cP [|\sum^N_{n = 1} a_n r_n|\geq c^{-1} \alpha], \] where \(r_n\) are Rademacher functions, hold for all \(a_1, \dots, a_N\) from a Banach space \(B\), and all \(\alpha > 0\) (i.e. the distribution functions of Sidon and Rademacher series are equivalent). An analogue of this result in the setting of noncommutative groups is also stated.