Starting with the two standard model of randomized communication complexity, we study the communication complexity of functions when the protocol has access to a defective source of randomness. Specifically, we consider both the public-randomness and private-randomness cases, while replacing the commonly postulated perfect randomness with distributions over $\ell$ bit strings that have min-entropy at least $k\leq\ell$. We present general upper and lower bounds on the communication complexity in these cases, where the bounds are typically linear in $\ell-k$ and also depend on the size of the fooling set for the function being computed and on its standard randomized complexity.