Mary Beard twitter and facebook

A lady called Mary Beard (who I've never met) is suffering a lot of abuse on twitter, the poor woman has received rape and death threats among other things.  Now nobody should have to put up with this, but it seems the Internet is left open for people to abuse each other with little recourse.
Frankly I don't do twitter, I did set up an account, I even made two posts... the first was something like 'this is my first post'  the second was 'I'm making a cup of tea'  Then I sat down and realised what a load of crap that was and why in the world would anyone want to bother reading it?
Face book I did for longer, I could sort of see the point, you could tell your mates you'd be down the pub at 8 o clock, or whatever (it also tell the buglers your out) post your pics for your mates or family to see etc etc.  But and it's a big but, the big problem with both twitter and face book is that other people can write on your page, this leaves it open to abuse.
Now I had something of a face book expert set mine up, so all the privacy setting were just right, but I still had people turn up as friends I'd never heard of and not accepted, but they got past the so called privacy settings, the same with total strangers writing on my wall.  It was a farce, there was bugger all private about it either, I only wanted my friends to see what I was posting but it seems anybody with a computer can see whats happening.  I gave up and closed it down,  I feel for poor Mary, she's trying to share with her friends and getting the moron trolls as well.