It is happening in the West. To take Australia as an example, on average one woman is killed by her partner or ex approximately every week. In my view, this counts as "problem in the West".
It shouldn't be surprising that an activist of any persuasion would be largely concerned with what is happening in their own country. Not only is it likely to affect them, or their friends, personally but it's also the place where they probably have the most chance of getting results. This doesn't mean that they have no interest in what is happening in other countries.
And whatever you may think of Sarkesian, AFAIK (not having read her stuff) the point that she is basically making is that if you are living in a society where women are regularly killed by their partners, providing "entertainment" which treats similar killings as acceptable/admirable/funny is not likely to help the situation, and in fact may say quite a bit more about the society than some people would like it to. On the face of it, this seems reasonable.
Yes, intimate partner and domestic violence is a problem (and usually relates to what I said earlier about women not being the property of a man). But it's not one of the problems that is being addressed by those modern, western feminists. Instead they focus on things that aren't actual problems. Like the "casting off my womb" art exhibit. What problem is that supposed to solve? Or Anita Sarkesian saying she spent "hours" playing video games, and people trolled her and so The Magic Patriarchy is out to silence her and she needs your money. Or whoever that annoying redhead one is that screams all the time.
With regards to Sarkesian just being anti-violence in video games, the video games she lamblasted didn't show intimate partner violence, they showed violence against random people; and they all showed far and away more men than women being the victims of violence. Women on TV don't die in gory ways, for the most part, because that would hurt the viewing public's sensibilities. And video games where women are supposed to be protected and rescued? That's sexism too, because women shouldn't be seen as "damsels in distress." So women can't be hurt, and they also can't be protected.
Here is a list of some of the ways that the modern feminist movement is just plain self-contradictory, including these ones from the comments:
11. We believe that all women should be able to choose for themselves what they wish to do in life, but if a woman chooses something considered "woman's work", we will let her know that she has failed as a "feminist" by making that choice.
12. We're sex positive, but we think all men are filthy rapey beasts who should cross the road to avoid us. And heaven forbid they should express or even think sexual thoughts about us, because that objectifies and dehumanizes us, and infects us with their beastliness.
http://www.skepticink.com/skepticallyleft/2013/01/06/dear-modern-feminists-so-this-is-what-youre-saying/
Topic: An apology from a Man