The rumors are frequent private meetings-- No, meetings themselves are not rumours; rumours are 'unsubstantiated news' and meetings are 'gatherings of people'. The rumours are of frequent private meetings -- Yes. 'Rumours' means 'unsubtantiated news' and 'of' means 'about': news about meetings .
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When the noun has to be augmented with a describing quality we can use 'of' before the noun in the sentence.Whether that is true or not, I am unsure, but that is irrelevant to this case.
'Too much of a coward' is an emphasised form of the other phrase.Obviously because of 'too much', but that does not explain 'of', which was what was being question