English Translation
How many people do you know well?
Couldn’t this also mean “how many people know you well”?
Indeed it can! 
The polite “Sie” can sometimes be ambiguous.
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I’m not sure how to disambiguate this, even if I try to make the sentence more explicit:
Wie viele Menschen gibt es, die Sie gut kennen?
Was ist die Zahl von Menschen, die Sie gut kennen?
These could still be understood either way! Is there some trick that Germans use to make it more obvious which meaning is intended? This isn’t a problem in the informal because “du” changes to “dich”, and the verb conjugation adjusts also, but “Sie” could be either nominativ or akkusativ, and the verb conjugation doesn’t help when speaking in the plural.
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In the very unlikely case that the context is not clear, you might change to passive:
“Von wie vielen Menschen werden Sie gut gekannt?”
Not exactly elegant, but unambiguous.
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Usually, by stressing the words differently.
Also, since there are only two meanings—if the question was misunderstood—simply saying that’s not what you meant and repeating the question, makes it clear to the other person that you meant the other meaning.