I've heard this a lot lately. Someone will say "Do you want to go with"? I can't help but think this is incorrect, that it should be "Do you want to go with me" or "us"? It seems wrong to end a sentence with a preposition, let alone end the sentence without a link to the preposition. Please, correct if I am wrong. This is driving me crazy!

It's known as pronoun dropping.

Linguistically English is classed as an NPD (non-pro-drop) language, so technically pronouns should not be omitted. However, as you mention, in everyday speech (or in certain dialects) pronouns will be dropped in various ways. There's not much you can do about it other than correct the person and tell them that English is non-pro-drop!

Note that there are a number of pro-drop (PD) languages, like Chinese or Japanese, where pronouns can be left out because they can be inferred.