For (Formula presented.) a Rademacher or Steinhaus random multiplicative function, we consider the random polynomials (Formula presented.) and show that the (Formula presented.) th moments on the unit circle (Formula presented.) tend to Gaussian moments in the sense of mean-square convergence, uniformly for (Formula presented.), but that in contrast to the case of independent and identically distributed coefficients, this behavior does not persist for (Formula presented.) much larger. We use these estimates to (i) give a proof of an almost sure Salem–Zygmund type central limit theorem for (Formula presented.), previously obtained in unpublished work of Harper by different methods, and (ii) show that asymptotically almost surely (Formula presented.) for all (Formula presented.).