Is multiplicative secret sharing secure?
Assuming that $p$ is prime, then you are in a cyclic group. Consequently, this is identical to considering the shares $s_i$ as "exponents" of a generator $g$ of $Z_p^*$.

Now we can write: $s_1 = g^{s'_1}, \ldots,s_{k}=g^{s'_{k}}$ and $s=\prod_{i=1}^{k} s_i$

Or we can view this as: $s = g^{\sum_{i=1}^{k} s'_i}$. 

Consequently it looks like a perfect (= information theoretically secure) additive secret sharing scheme "in the exponent" to me.