Native speakers use them. In fact, the example is likely a paraphrase of Sleeping in my orchard, a serpent stung me. Shakespeare (Hamlet, Act 1, sc 5). Shakespeare was a native speaker.
Most listeners would have no trouble knowing it was me who was sitting. This is usually more obviously true when we hear the sentence in its context. Sentences occur in contexts. Still, it is very probably not the best way to construct the sentence because, although a listener would probably not have trouble understanding who is sitting, the language is fairly likely to cause their mind to need to ask itself who is sitting. Although the answer will likely come quickly, better writing style avoids causing such distraction in the first place.
So native speakers are fairly unlikely to use this particular construction unless they wish to achieve a special effect, and unless the context (what's established in the situation and preceding language) reduces the chance of such a distraction occurring.
I don't think English learners are particularly likely to produce such sentences either!
Some will insist that "dangling modifiers" are always grammatical errors as violations of syntactical rules. However, this can only be a matter of prescriptivist opinion or taste.
Pullum, a linguist and co-author of the excellent Cambridge Grammar of the English Language, gives several examples of "dangling modifiers" that "are happily used by even expert and careful writers:
Speaking of sales, what do the third quarter figures look like?
Moving right along, this slide shows the third quarter figures.
Seeing as you're here, there must be some ulterior motive for the visit.
Without detracting from Fowler's point, aren't some Anglo-Saxon verbs overused?"
It makes more sense to label such sentences bad writing when they are likely to cause confusion or are misleading, or would just sound odd or nonsensical:
Feeling sleepy, the bed was put to use.
One might like to call this ungrammatical, but it is difficult to cite a formal rule that is broken, when we have just seen that other sentences that break the same rule are perfectly acceptable.
If this is the topic of a lesson in an English class or textbook, I submit it's a bad one.
People who simply read and hear enough good English that they can understand are unlikely to produce such sentences, though they cannot explain which "rules" they are following.
As Pullum ends his article:
If ever someone tells you that the rules of English grammar are simple and logical and you should just learn them and obey them, walk away, because you're getting advice from a fool.