The most well-known construction of a non-measurable set is the Vitali set. The idea behind Vitali sets is to split up the space (such as $[0,1]$) into equal-sized copies (guaranteed by translation invariance), by looking at something like $\mathbb{R}/\mathbb{Q}$. This same idea is used in "Visualizing a Nonmeasurable Set" to construct non-measurable sets on the torus.

Another construction I know about is on the probability space of infinitely many coin tosses $\{ 0, 1 \}^\mathbb{N}$. In this case, instead of modding out by $\mathbb{Q}$, you can mod out by "switching the outcome of finitely many coins". This approach is taken in these probability lecture notes.

All of these constructions seem closely related: each time, we have a way to decompose our set into "translation-invariant" copies. **My question is how these sorts of constructions of non-measurable sets generalize.** In the Wikipedia article on Haar measure I read:

Unless $G$ is a discrete group, it is impossible to define a countably additive left-invariant regular measure on all subsets of $G$, assuming the axiom of choice, according to the theory of non-measurable sets.

This seems very close to an answer to my question, but the Wikipedia article doesn't elaborate here. So, how does the construction of non-measurable sets on a non-discrete group $G$ work? Is the general intuition similar to the examples I've described here? What happens for discrete groups (could this be related to amenable groups, which I know you can put measures on)?