How does the author's native language impact the likelihood of having his publication accepted in an English-speaking venue?
It's probably more a commentary than an answer. I look forward to others' input on this issue as well.

As an ESL myself, I joined the English academic community knowing that language will be an extra barrier. I wrote (or perhaps still write) many awkward sentences that are grammatically sensible but funny to read from the angle of general English usage. To compensate this shortcoming, I read papers and study their styles, I read books on scientific writing, and I coauthor with native speakers. My English has improved somewhat and now I can even throw a couple trivia about academic writing in front of my native speaker colleagues. But deep down, I know it's a never-ending project, and chance is I will not be as fluent as a native speaker, and should perhaps settle for "good enough."

If I were to talk to your author friends, I'd probably ask if those submissions were anonymous. And if they are anonymous, and in multiple occasions comments like "English needs improvement" were given, I would suggest them to spend some time to work on their English rather than complain about unfairness. Non-native speakers having more difficulty to publish in English speaking channels is logical; to expect a fair ground that would totally ignore linguistic difference is unrealistic.

To formally answer the question, I have not come across with any study similar to what you described. And even there is one, I will study the results very cautiously. Native language cannot be randomized, so it's impossible to have a randomized controlled trial. This means what we will likely to see are observational studies, which are prone to missing important confounding variables: Native language intertwines with too many factors (education system, country of origin, culture, socioeconomic status, writing style and syntax construction, quality of English education, existence of equivalent concepts in English, etc.) that probably cannot be fully adjusted for in the analysis. At the end, we will likely see that non-native speakers are at a disadvantage, but the true cause of such disadvantage may be lurking elsewhere, and the true effect contributed by the linguistic difference alone is next to impossible to be parsed out.

Truth be told, I believe discrimination exists. If a reviewer reads that the data were collected in a non-English-speaking country, he/she may pay more attention to the grammar, or be more likely to suggest comments like "Please consult an editor or a native speaker in order to polish the English." I am fortunate enough to be working in the biomedical field in which most writings tend to follow a framework of being succinct, direct, and optimally simple. Most of the time I could easily parse out the ideas from the writing quality and give comments separately to both. Researchers in other fields may feel differently.