DISCLAIMER: I have a Y-chromosome. I have a penis. And there are some things I will just never experience and never know. However, I think our similarities massively outnumber our differences. So I will muster on.

The first point is crucial. Both sides in the debate are screaming past each other, never really addressing the concerns of the other side or establishing any common ground. It has devolved into a nonsensical, circular national argument that has given ample room for moral posturing as well as political manipulation. In short, we need to basically start from scratch to make any sort of progress on this front.
The problem is that the two sides interpret the issue within two different narratives. For the pro-choice movement, abortion is part of the long march towards women's rights. Women deserve every opportunity in the world, including the right to an education, a career, and a family on their own terms, not simply their husbands. In this way, the choice of having an abortion is one more liberty granted to women, symbolizing full citizenship. When a bunch of men start extolling the ethics of the situation, many women recoil and say, "Listen, don't tell me what to do. Don't impose your ideology on me. I have the right to choose on my own."
For the pro-life, they have to interrupt at some point, and rightfully ask, But what about the baby? To them, abortion is so clearly an ethical issue, a question of morality, a question of the very sanctity (read: preciousness) of human life. And any other frame of reference, especially something so pithy as a political story, is so insignificant. It falls by the wayside. Susan B. Anthony and Mary Wollstonecraft have nothing to do with it. There are two simple questions -- Is it alive? Is it human? If the answer is yes, then abortion is the depravation of human life. Then it is akin to slavery. Then it is akin to murder. And opposing it is an ontological imperative.
Notice, the images these stories evoke are so drastically different. In one, the proud and forward-looking woman; in the other, the helpless fetus drawn to slaughter. Both sides have sufficiently demonized the other.
But, like two children arguing over who gets to be quarterback, they forget they are on the same side. They both hate the situation of abortion. They hate the moral anguish, they hate the lack of support, they hate that pregnant women are driven to abortion in the first place. They hate the public debate about private lives. But they confusing motives in the other side.
We can reduce unwanted pregnancies through encouraging adoption, through supporting pregnant women, by making pregnancy socially acceptable, by teaching about contraception honestly, by destigmatizing sexuality in our culture. When it comes to the policy goal, only common ground is useful to work off of, so this is what both sides share.

But on the personal hypothetical decision, like so many of my pro-choice friends, I am pro-life. I cannot imagine contemplating the fusion of 23 of my chromosomes with another human being's and not feel a sense of awe and wonder and immediate obligation. I believe experiencing the creation of life for the first time would make the thought of abortion unconsciable to me in that moment. I would say to myself, "Not everything in life goes as planned." And I would feel compelled to reorient my life accordingly. Making any other decision would leave me with a deep, burning sensation of moral shame, for being so selfish. This is what pro-lifers cannot see beyond in the debate. Their moral frame of reference is so intensely fixed in place that other, more cosmopolitan frames are unconsciable.
So personally, I know the beauty of life, and I can imagine the shame of immorality. Publicly, I just want both sides to stop acting like they're going to kill each other.