I as a tutor, and often have to explain this to a student (usual way is to take away the other noun/pronoun and ask them, “Would you really say 'Me had a sandwich`?”).
But pretty much everywhere I go on da interwebz I see the former almost exclusively. Is the latter construction doomed in the long term?
[Saw another pronoun topic here this week, but if this is more IMHO material won’t quibble.]
Like most things, it’ll go both ways.
As with most linguistic events, future researchers will record a variety of usages during discourse in different venues and different levels of formality.
I see errors that go the opposite way much more frequently. “Please send your response to Joe and I.” It is usually done by people trying to sound smart.
Nope. Children overuse “me and X” until the point when they listen to old people absolutely hammer into them the idea that it’s always wrong, at which point adults who should goddamn know better overuse “X and I.” “Jamie gave Morgan and I a sandwich.” I hear it all the time.
Overcorrection is much more aggravating than being regular wrong.
English grammarians had a phase in which they tried to make English grammar more closely resemble Latin grammar and predicate logic. But natural languages don’t have to follow the same systems as each other, and certainly not follow constructed languages.
“Me and Jane rode horses” has been and is correct English in many speech communities. It is not a formal register, but a language is defined by all of its speakers, not only the elite. In contrast, “me rode a horse” has never been widely accepted. They are separate constructions–there’s no reason that a natural language should be consistent across different circumstances.
Humans have a strong tendency to exclude others based on superficial differences, like language usage, and so you’ll often find strong reactions to differing registers and dialects.
Steven Pinker makes the point that there’s no legitimate grammatical rule that says that the conjoined components in a noun phrase must agree with the case of the phrase itself. The form “me and X” is acceptable at least in informal use by virtue of its being such common usage and the aforementioned absence of any rule to the contrary.
One could use the same argument to justify “Please send your response to Joe and I” except that it’s obviously just a hypercorrection based on a failure to understand the prescriptivist version of the supposed rule about using subjective case pronouns in subjective noun phrases. The artificial hypercorrection is why it sounds so odd.
I will gladly endure “Me and him had a sandwich” for the rest of my life, if I could avoid ever hearing “That’s between you and I.”
ETA: what Wolfpup just said.
Very true. In fact, in French, the “me and …” construction is completely correct - Je mange le steak (I am eating steak), but Moi et Robert mageant le steak (Me and Robert are eating steak).
Me too. Or is that “I too”?
I also…
Yes, in fact, if you’re going to follow the rules of pedantic prescriptivism, it definitely should be “I, too”! Because after all, technically what you’re really saying is “I, too, see errors that go the opposite way …” 
The downside is that if you wave your hand in an elementary school classroom and confirm your agreement with something by claiming “I, too!”, there needs to be a considered balance between this inarguable pedantic correctness and the risk that after class some of the larger boys will hold you upside down and dunk your head in the toilet bowl. IOW, there is “formally correct proper English” and there is “being a dork”. And unfortunately in the middle of all this there is also “being an illiterate buffoon”.
My observation is that the wrong/awkward use of ‘I’ instead of ‘me’ is mostly done by Americans who are mistakenly thinking that it is correct. ‘Me’ is rarely wrongly used.
They’re both doomed. Not knowing the latter formulation but convinced that the former is wrong, people are telling us that “MYSELF and X” had a sandwich. Or that a sandwich was ordered “by X and MYSELF”.
visceral shudder
The English language is gonna do what it’s gonna do, with usage determined by teenagers in California. I shall deal with the contretemps by repairing to the library with port and cigar and clucking over the decline of the word and world.
Demanding that these constructions are the only correct ones was the invention of a bunch of 19th century elitists who were terrified by the rising literacy of the lower classes. They were pretty much the same people who came up with social darwinism and eugenics.
We need to stop this train the way the others were made unacceptable in decent society.
That’s NOT the same thing as saying anything goes. People started recognizing and decrying these artificial distinctions a century ago. And the language is not in any way doomed, destroyed, or degraded. Proper language is still the norm in proper situations. Casual language is the norm in casual situations. True illiterates get called out all the time.
Shall and will and whom and who are the only distinctions that come to mind as having been truly doomed over the last century. I don’t miss them.
Leaving your lawn completely unprotected.
I once heard of the perfect inversion: two teenage girls in some very rural part of England, discussing another- “Why do 'er wave at we? Us don’t know she!”
Try watching an Australian soap opera. It seems to be universal there. And not unknown here in the UK.
There’s a Norwegian dialect area that has Us replace We (More precisely the Norwegian “Oss” replaces “Vi”). When the speakers of that dialect explain this feature it becomes: “Us say us us.”
(Repeating the subject at the end of a statement like that for emphasis is a general feature in Norwegian.)
Have installed claymore mines