The stickler, encountering a statement such as "Me and Madison are going to the mall," will usually start to fret about the failure of public education, the decline of literacy, and the descent into barbarism. Expressions such as "nails on the chalkboard"* will commonly be used.
I would like to suggest, in the kindest and gentlest way, that this reaction betrays a lack of sophistication about language.
Merriam-Webster's Dictionary of English Usage notes that the "me and someone" pattern is a speech form, usually associated with the speech of children. That is, it is a common pattern of usage that can persist into the speech of adults. A 2011 post at Language Log by Mark Liberman examines the patterns of usage in corpora and discovers that "me and someone" is a common construction, while "I and someone" almost never appears.
What this suggests is that "me and someone" is not an error. It is merely a common form of non-standard English—though a strongly stigmatized one, as the Language Log post notes. "I and someone" would be an error, a pattern of usage that almost never occurs. (That post also includes theories about the prevalence of "coordinate accusatives corresponding to non-coordinate nominatives." Worth a read.)
To insist that "me and someone" is an error, then, is to assume that standard English, one dialect of the language, is the only correct one and that anything varying from it must be wrong. If you think so, this assumption is causing needless pain and cluttering up the internet with pointless fulminations.
Let me suggest that abandoning this mistaken belief can prove liberating. Once you discard "non-standard=error," you can allow people casual conversation and informal communication without discomfort. People who use "me and someone" or double negatives or irregardless will cease to trouble your slumber. You will be able to substitute some other hobby for your custom of scolding people—perhaps something offering healthful exercise.
It would mean, of course, that you would have to find some other means of affirming your elevated status, but human beings are remarkably inventive at finding grounds—dress, grooming, choice of automobile, tastes in music, support of sports teams—for some imagined superiority. Go for it.
*Nails on the chalkboard" and similar expressions should be understood thus: "I am employing an inconsequential personal irritation to publicly mark my intellectual/social/moral superiority."