I'll preface this by saying that English is not my mother language and I'm sorry if this isn't the right community, but I didn't find a more appropriate one.
Last year I started to notice more and more people on YouTube for example using the verb "to put" without a preposition -- like "Now I put the cheese" -- which sounds very weird and kind of feels wrong to me. Is this really used in spoken English and is it grammatically correct?
No for most people because "put" would require two complements - the object and the place. (It's a lot like a ditransitive verb, even not being one.) And in this case the place is missing, you'd expect it to be at most replaced with an adverb (e.g. "now I put the cheese there").
The exception I can think of as a native speaker is when the place is strongly implied. Like, "Now I put this..." and trail off, but it's obvious where I'm putting it because you're watching me put it somewhere that I've probably indicated multiple times already.
Caveat lector: I'm not a native speaker, so take what I'm saying with a grain of salt.
This exception seems to be a case of right-edge deletion - you're generating the sentence as if you were to utter the adverbial/prepositional complement, but then you chop the sentence just before the element. This can be shown by the following:
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