Recruitment dynamics, or the distribution of the number of offspring among individuals, is central for understanding ecology and evolution. Sweepstakes reproduction (heavy right-tailed offspring number distribution) is central for understanding the ecology and evolution of highly fecund natural populations. Sweepstakes reproduction can induce jumps in type frequencies and multiple mergers in gene genealogies of sampled gene copies. We take sweepstakes reproduction to be skewed offspring number distribution due to mechanisms not involving natural selection, such as in chance matching of broadcast spawning with favourable environmental conditions. Here, we consider population genetic models of sweepstakes reproduction in a diploid panmictic populations absent selfing and evolving in a random environment. Our main results are {\it (i)} continuous-time Beta and Poisson-Dirichlet coalescents, when combining the results the skewness parameter $α$ of the Beta-coalescent ranges from $0$ to $2$, and the Beta-coalescents may be incomplete due to an upper bound on the number of potential offspring produced by any pair of parents; {\it (ii)} in large populations time is measured in units proportional to either $N/\log N$ or $N$ generations (where $2N$ is the population size when constant); {\it (iii)} it follows that incorporating population size changes leads to time-changed coalescents with the time-change independent of $α$; {\it (iv)} using simulations we show that the ancestral process is not well approximated by the corresponding coalescent (as measured through certain functionals of the processes); {\it (v)} whenever the skewness of the offspring number distribution is increased the conditional (conditioned on the population ancestry) and the unconditional ancestral processes are not in good agreement.