Yes, this is fairly common in casual speech, not so common in formal writing.
Note that often this indicates that the speaker considers the statement surprising. In general, if you were, asking a person if they wanted to remain in their present location, you would use the "by the book" construction, "Do you want to stay here?" But if a person indicated they wanted to remain somewhere that you thought undesirable, like some place uncomfortable, or if you expected that they would want to leave, perhaps you thought they were in a hurry to move on, then when the person says, "No, I think I'll stay," you might reply in surprise, "You want to stay here?"
It is used as an alternative to a question in very informal speech. Instead of, "Would you like to stay here, sir?" one might ask, "Ya wanna stay here, bud?" This is especially true as the construction of the sentence becomes more complex. "You guys spoke to anyone?" is not unlikely, because the more formal statement would be, "Have you gentlemen spoken to anyone?", which is several levels higher in formality.
Love this question. However, it can happen with sentences as well as questions (much like I did in the first sentence of this comment). When I catch myself doing this in writing, I usually fix it by adding the omitted verb or pronoun. I think that it's usually best to restrict this form to conversational use, and to avoid it in writing. – J.R. – 2013-09-03T01:53:44.013
@J.R. It's easy to explain the first example as omission, but not the second. The second can't have a deleted auxiliary such as have or did because the main verb (spoke) is already tensed. The OP's explanation (using statements as questions) works, though. – snailplane – 2013-09-03T02:17:38.307