In this paper we obtain (almost) optimal results concerning randomly weighted one-sided ergodic Hilbert transforms. Given an iid sequence of centered random variables (Xn) in L log L, we show that there exists a universal set $\Omega'$ of probability 1 such that for any $\omega\in \Omega'$ the realization $(X_n(\omega))$ is good for the almost everywhere convergence of the weighted one-sided ergodic Hilbert transform associated with any dynamical system and any $g\in L\log L$. The method applies to powers along subsequences with ‘small’ growth and when considering Dunford–Schwartz operators instead of pointwise transformation. If the (Xn) are symmetric, but only in L log log L, we obtain a slightly weaker result.