For $X(n)$ a Rademacher or Steinhaus random multiplicative function, we consider the random polynomials $$ P_N(θ) = \frac1{\sqrt{N}} \sum_{n\leq N} X(n) e(nθ), $$ and show that the $2k$-th moments on the unit circle $$ \int_0^1 \big| P_N(θ) \big|^{2k}\, dθ$$ tend to Gaussian moments in the sense of mean-square convergence, uniformly for $k \ll (\log N / \log \log N)^{1/3}$, but that in contrast to the case of i.i.d. coefficients, this behavior does not persist for $k$ much larger. We use these estimates to (i) give a proof of an almost sure Salem-Zygmund type central limit theorem for $P_N(θ)$, previously obtained in unpublished work of Harper by different methods, and (ii) show that asymptotically almost surely $$ (\log N)^{1/6 - \varepsilon} \ll \max_θ|P_N(θ)| \ll \exp((\log N)^{1/2+\varepsilon}), $$ for all $\varepsilon > 0$.