NOTE: This doesn't answer the question... which was how to multiply two secrets...
I played around with it (empirically) and it looks like you can multiply or add as much as you'd like to the secret and the resulting combination is also multiplied/added by the same amount.
IE: I multiplied the share data segment in modulo P, and got back the secret multiplied by the same number.
Of course, this makes sense because you are essentially multiplying the Y-values of the line used to derive the secret.
Thus m parties can multiply and add to their shares, and while none of them will know the secret, the result of combining those shares will be a consistently modified secret.
Inconsistencies in the modified result can be detected as well using VSSS (verifiable secret sharing), and altering the verification generator.