Impact of a non-uniform distribution in a secret sharing scheme When it comes to the secret sharing schemes, it seems that the random numbers need to be drawn uniformly. Is there any difference between the random numbers from uniform distribution and the non-uniform distribution? Do different distributions affect the security of the 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.