The title goes for both extremes on the curve, that means on one hand the TRP/MGTOW/Extreme parts of the manosphere and on the other hand radical feminists/Tumblr SJWs. But since on this sub, we don't seem to have much of the latter part (really I've very rarely seen anyone here argue that men have so much more privilige than women. People that think this definitely exist, but not much on this sub), I'm going to use mostly examples from the TRP side. Feel free to swap "men" and "women", to subtitute "TRP" with "SJWs" and you can see the argument from the other side.

In my opinion, those that argue that the other gender has all the privilege (a popular quote from TRP: "Western women are the most privileged humans ever), take what they have for granted, and look at what they want and call those who have it privileged. From the perspective of the man, this means that having sex relatively safe (no risk of pregnancy, STDs, little chance of getting raped since most men are stronger than women, easier to get off during sex) are default traits, and him having them does not mean he's privileged in his mind.

But the things he can't do and women can (ie getting sex easier, better pickings) but that he wants to do, are examples of female privilege. But in calling out this privilege, he totally ignores the downsides that come with it. In my view, that constitutes a lack of empathy, because they cannot look past their own perspective at all.

Another good example is "women don't fall into poverty/have less chance to be homeless than men". That is indeed an advantage of being a women, and men recognize that. What they don't recognize is that it's easier for men to climb higher on the social ladder, and that they are taken more seriously in leadership positions in general. Again, they look at the good things the other gender can do, but ignore the bad that comes with it and come to the conclusion that the other gender is priviliged over them.

On the other side, Tumblr SJW's will flip shit about men who get into good high paying leadership positions easier, but ignore that men have less social structures to support them when the chips are down. They get angry that men are stronger than women and can use this "privilege" to rape, but don't acknowlegde that there is very little support for male rape victims. Again, that, to me is a total lack of empathy.

So, in my opinion, the extremes on both sides have a problem in their reasoning: they lack the empathy to see that the other side may also have things that are not cool to deal with. They only see the cool things they don't have and call privilege in response.