A couple of points, and maybe I’m being too picky; although many of the pieces in the collection may be from the 1950’s, this particular one has publication information shown on the first page that says it was published in 1963; Strange Suspense Stories #66.
That being said, if I recall correctly, it’s an awfully lot like Zenna Henderson’s first published short story from 1951; “Subcommittee” (Magazine of F&SF). This story involves negotiations between humans and aliens, while the children are playing together unbeknownst to the adults (crawling under a fence to play, if I’m remembering correctly), and then the wives meeting and becoming friends. All the while, negotiations are breaking down between the husbands, who are the chief negotiators.
Still though, Henderson’s audience was a different one than this was intended to reach, and it’s a laudable message,
