The joint distribution of sequences $(f_\ell(P_\ell(n)))_{n\in\Bbb N},\ell=1,2,\dots,d$ and $(f_\ell(P_\ell(p)))_{p\in\Bbb P}$ respectively, where $f_\ell$ are $q_\ell$-additive functions and $P_\ell$ polynomials with integer coefficients, is considered. A central limit theorem is proved for a larger class of $q_\ell$ and $P_\ell$ than by Drmota~[3]. In particular, the joint limit distribution of the sum-of-digits functions $s_{q_1}(n),s_{q_2}(n)$ is obtained for arbitrary integers $q_1,q_2$. For strongly $q$-additive functions with respect to the same $q$, a central limit theorem is proved for arbitrary polynomials $P_\ell$ with the help of a joint representation of the digits of $P_\ell(n)$ by a Markov chain.