Secret sharing scheme verifiability
I am fairly certain, that your proposed secret sharing scheme is actually not secure in any way:

If a party receives $s_i(x)$, then the secret polynomial is of the form $f(x)=s_i(x)+m_i(x) \cdot r(x)$ for some polynomial $r(x)$. And that is not equivalent to all possible polynomials. Even worse: If you allow the parties to choose $m_i$ themselves and of arbitrary degree, the parties can choose it do that they basically get $f(x)$ in the clear. 

Example: $f(x)$ has degree $d$, then $m_i(x)=x^{d+1}$ gives back $f(x)$. Even if the degree is restricted, not all polynomials are possible.

But even if we assume, the secret sharing was secure, then your proposed algorithm would break it:

If the party can verify if their share was created properly, then the party can use this very procedure to learn something about $f$ instead. For any secret sharing scheme, giving the parties additional knowledge on top of their individual share will most likely break the information theoretic security (and otherwise it has to be proven). And since you have no computationally hard problem at all in your construction, it can't even have computational security.