Yes, people are sharing fake news through Facebook and Twitter. People are increasingly using social media platforms to receive news through a “filter bubble,” where they will only end up reading the news and opinions they already agree with, regardless of whether those “facts” are actually true. In the long run, that will be poisonous to our nation, convincing everyone that their own opinions are infallible and the opposition is at best stupid or at worst evil.
I shouldn't have to say it but this applies to everyone. We've had media for forever but the increasing polarization of politics may be related to the rise of these particular social media platforms. The timing is certainly coincident.
I have specifically hidden many fake news sites (both left and right) from my Facebook news feed and despite this, it seems like I am seeing fake news on a daily basis. Images containing a bunch of "alternative facts" are another issue. I am smart enough to differentiate between truth and reality, but I have serious doubts as to whether many other Americans are. I realize that reputable news sites still get far more visits, but I see fake news articles being shared with troubling regularity. My group of Facebook friends generally skews better educated and more left-leaning than the general public, although I see a high correlation between people of lower educational achievement and the propensity to share fake news articles (on both the left and the right). In the end, we will never know how much fake news impacted the outcome of the election, but I suspect it was not insignificant or minor.