Set the parameters of a Erdos-Renyi graph generator to get a specific mean degree
Average degree and mean degree are the same. In the $G(n,m)$ model, the average degree is $2m/n$. In the $G(n,p)$ model, the expected average degree is $np$. The actual average degree has normal distribution with mean $np$ and standard deviation $\sqrt{2(1-\tfrac{1}{n})p(1-p)}$, so it is pretty close to $np$ with high probability.

When $p=c/n$ for fixed $c$, the degree of a single vertex has roughly Poisson distribution (with mean $c$); formally, as $n\to\infty$ the distribution tends to Poisson. In practice, this means that for "small" $c$ and "large" $n$, the degree will be Poisson. Furthermore, while there is some dependency between vertices, we still expect the entire empirical degree distribution to be close to the same Poisson distribution (i.e., we expect the fraction of coordinates having degree $d$ to be roughly $e^{-c} c^d/d!$); this is probably proved formally in the literature. 

When $p$ is constant, the degree of a single vertex has roughly Gaussian distribution, and it is probably the case that the joint distribution is also close to the appropriate multivariate Gaussian.