Do you want to come with?
Can I come with?
I seem to hear this construction more often in recent years, but it still grates on my ear.
I know it's often said that one shouldn't end a sentence with a preposition, but I class that rule along with split infinitives (i.e. — pedantic tosh).
Is it always informal/casual speech to end a sentence with "with"? Was it ever thus?
I’ve heard of it, but never heard it. The absence of a direct object after a phrasal verb that normally has one is not unprecedented. FF has already mentioned come after, but there are other examples. You can get on or get off a bus, or you can just get on or get off. You can go without dinner, or you can just go without. I don’t imagine anyone objects to those now, if they ever did. I suspect the only thing intransitive come with can be charged with is being new.
I wouldn’t be surprised if, as Lunivore says, it’s a construction found in Afrikaans. German has the cognate separable verb mitkommen (although admittedly that occurs only intransitively).
It's perfectly fine to end a sentence with a preposition, don't believe any fear-mongering to make you believe otherwise.
Sometimes it may sound awkward, sometimes it's more suitable for informal writing, but there is no prohibition against it.
Having to read comics about grammar jokes, I guess that's something I could live with.
However, the examples you cited don't sound like correct grammar to me. It's not because they end with with, it's the fact that they omit the personal pronoun:
Do you want to come with us?
Can I come with you?
They're informal speech patterns, and yes, there are grating, kind of like asking "How are you?" and getting "Thanks, I'm doing good."
"Verb + with + [nothing]" is a dialect feature that is widely accepted in America. The Yale Grammatical Diversity Project (YGDP) has a great article about it. (This is different than the inversion we see in questions and similar constructs, where there is a proper prepositional phrase instead of "nothing": "Who will you go with?" => "go with who". For that, see When is it appropriate to end a sentence in a preposition?)
In sentences like "Can I come with?" context determines the exact meaning: Who's leaving? That's who the speaker wants to go with.
There are at least 6 verbs that can work like this, mentioned by Spartz (2008): come, go, ride, bring, take, and carry; a listserv suggests that be is also possible ("A friend was with and she drove me home"). Not all verbs are equally acceptable, however. Consider this map (created by Jim Wood on Oct. 11th, 2019, based on YGDP's data):
See also the YGDP interactive map.
It's most common for "come with" and "go with" to sound natural. (They both sound natural to me.) Those in a dark green area will find more verbs acceptable. Spartz (mentioned above) found all those verbs while researching Minnesota, which is entirely dark green.
The origin is likely Germanic, according to YGDP:
Many people have identified similar constructions in other Germanic languages, including the Scandinavian languages, as the source of this phenomenon. This would explain its presence in places like Wisconsin and Minnesota, where many Norwegian, Swedish, and German speakers settled.
Lunenburg English, an area that has significant German influence (but is too Canadian to be in the YGDP data), also uses "with" like that: "Will you go with? I am going with. Come on with!" (examples from Wikipedia).
