I'd also wonder what you are giving her. The cat I knew who begged most for human food was being given only hamburger and rice by her well-meaning owner, which was not a well balanced diet for a cat. So the cat begged. You might try one of the higher-quality veterinary diets to see if her food is the problem.
what are you giving her for cat food? I hope its canned and NOT dry....of course people food tastes much better that theirs. and to share some IMO isn't harmful...but the begging isn't appreciated. maybe feed her before you all eat so she is full, that helps....
as far as the chewing, maybe she is having tooth problems, have you checked her mouth to see if she has red gums or something going on there?