Impact of a non-uniform distribution in a secret sharing scheme
Just to add to the other answers, you may also want to take a look at leakage resistance secret sharing, for example this work. 
The idea is to study the following question: 


  What is leaked about the underlying secret when the adversary gets some leakage from the shares?


This turns out to be a very interesting question, and in some sense, changing the distribution can be seen as "leaking" information since some shares may become more likely than others.

Interestingly, in some cases, even a 1-bit leakage can completely break security.
For example, if you use Shamir's secret sharing over $GF(2^n)$, then leaking the first bit of a large enough amount of shares will leak the first bit of the secret! So if you have a distribution that generates shares for which it is more likely that the first bit is, say, $1$, then you will know something about the secret.