Distributions which are intractable to sample from? I'm looking for an interesting family of probability distributions $P$ that is intractable to efficiently sample from. 

I'm not sure what the right notion of intractable is, though I know the notion of efficiently samplable: the existence of a polynomial-time randomized Turing machine that can simulate samples from $P$. 


  Q1: Is a reasonable notion of intractable that if there exists such a randomized Turing machine, it can be derandomized to solve an NP-complete problem? If not, what is the right notion? Is there a good reference for this? (Or just some good search terms.)


There are a lot of tautological examples along the following lines: there is no algorithm to sample from [set of solutions to NP-complete problem union a symbol for no solution], because there is an obvious reduction to [that NP-complete problem] by taking the output of the sampler. 


  Q2: What are some more interesting examples of intractable sampling problems than these?


There are also examples perhaps like this: Given a set $X$, the goal is to sample uniformly from $P(P(X))$ (powerset). If you could do so with a polynomial time turing machine, you could produce exponentially many bits of entropy in only polynomially many coin flips, which is impossible (because $H(f(Y)) \leq H(Y)$ for any random variable $Y$ and deterministic function $f$). 

In order to avoid this kind of example, I want the underlying set of the distribution to grow at most like $2^{p(|x|)}$, where $p$ is a polynomial in the size of the input.
I'll expand my comment to an answer. Many combinatorial structures in graphs are actually NP-hard to sample from.

The earliest example I can think of is JVV86 (Thm 5.1), which shows that there is no polynomial-time algorithm to randomly generate a cycle in a directed graph, unless NP=RP.

Similar results along this line are NP-hardness for sampling independent sets, colourings etc. See e.g. DFJ02, Sly10, GSV15. Note that the corresponding search problems are all easy. 

The problem can also be defined by "soft" constraints. E.g., sampling from an anti-ferromagnetic Ising model is NP-hard. See JS93 (Thm 14).

Results of this sort are usually stated as NP-hard approximate counting problems, due to the standard self-reductions JVV86.