Clarity: the article is clear and well written, In this aspect the paper is an "accept" for me. (6) Significance: results are significant as they are new and they will generate some impact for practitioners. This is an accept as well (6) Quality: this paper is of high quality, it is clear there is a significant research effort behind. The combination "theoretical results + empirical validation in simple cases" is sensible given the type of paper this is, and the audience. Accept too (6) Originality: This is the item where I tend to reject more than to accept (5). I think it is definitely original, but all the theoretical contributions seem to me a bit marginal: I am very familiar with Bernton et al 2018, the paper that develops the technique (in turn, mainly based on Basseti et al 2006 and Pollard 1980) that is used here. After reading the supplement of this paper I am left with a "deja vu" feeling; some proofs look way too similar to Bernton et al, making me wonder whether they are rather straightforward adaptations. Indeed, It is no surprise all results of Bernton et al hold here too as this is a kind of average of one-dimensional wassersten distances. It is no surprise as well it is possible to establish a distributional limit in the multidimensional case here (unlike Bernton et al), although I think it is a very nice observation. Question for the authors: Does the obtained distributional limit here gives any intuition about what should be the limit in the multidimensional case for the vanilla wasserstein distance?

The paper is of high quality. It also comes with reasonable clarity. Overall it brings important new analysis to the important topic and the work seems to be original from what I can say.

***** After Author Response and Reviewer Discussions ***** I have gone through all the other reviews, the meta-reviewer's comment, and the authors' feedback. I will keep my evaluation unchanged. ********************************************************** *originality: To my best knowledge, the results are original. The methodologies of analysis belong to classical asymptotic statistics, but the problem analyzed is new. *quality: Due to the time constraint I did not go through the proofs. The claim in the abstract is well supported by the theorems. The work appears to be complete. The authors are honest about claiming the strength and weakness of their work. *clarity: The submission is clearly written, well written and of elegant style. Since this is a theoretical paper, the proofs provide enough information for an expert reader to varify the theorems which are the results. *significance: The results are important and significant for understanding the behavior of MSWE and MESWE. This paper is likely to be cited by people working on the application side of these two estimators. The problem is difficult and the authors have provided better analysis than that in literature to my best knowledge. The analysis provided advances the state of the art in a demonstrable way. The analysis is theoretically unique.