"John's teeth are being brushed by John" means, there are two Johns, like this:
John Smith's teeth are being brushed by John Sampson.
The noun John (Sampson) does not refer back to John (Smith). That is, they are not co-referencial: the two Johns are not one and the same person. The same holds true for the pronoun 'his':
His teeth are being brushed by him.
The pronouns 'his' and 'him' are not co-referent.
The teeth are being brushed by John.
'teeth' belong to the body, so the noun requires a possessive pronoun, either overt (written/spoken) or covert (unsaid/not written). In saying, "The teeth", the listener/reader assumes the teeth belong to something or someone, and given the passive structure, the listener/reader also assumes "the teeth" do not belong to John". We could modify the noun so as to show it refers to John, like this:
The teeth belonging to John are being brushed by John.
John's teeth are being brushed by John himself.
But, they're pretty odd things to say.