We determine the Krieger type of nonsingular Bernoulli actions $G curvearrowright prod_{g in G} ({0,1},mu_g)$. When $G$ is abelian, we do this for arbitrary marginal measures $mu_g$. We prove in particular that the action is never of type II$_infty$ if $G$ is abelian and not locally finite, answering Krengels question for $G = mathbb{Z}$. When $G$ is locally finite, we prove that type II$_infty$ does arise. For arbitrary countable groups, we assume that the marginal measures stay away from $0$ and $1$. When $G$ has only one end, we prove that the Krieger type is always I, II$_1$ or III$_1$. When $G$ has more than one end, we show that other types always arise. Finally, we solve the conjecture of [VW17] by proving that a group $G$ admits a Bernoulli action of type III$_1$ if and only if $G$ has nontrivial first $L^2$-cohomology.
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