I've learned a few things about chatting with people who vehemently dislike me and everything I hold true and precious. Let me share a few with you.
Do not defend yourself. You will be personally attacked, and it's going to hurt. Your natural response will be to defend, to cover, and even to retaliate. Fight this propensity as much as you can. Your opponent will be thrown off; also, she will be less likely to put up a wall of defense herself.
Do not defend yourself. The attacks are not personal. They sound personal, but they're not. Your opponent is trying to avoid conversing about the actual issue on which you disagree, and attacks on one's character usually work. Your opponent is also trying to verify the truth of his own conviction by painting you into a preconceived box, a stereotype of people like you. This stereotype allows you to be dismissed more easily. Be attacked; you can handle it.
Be prepared. This doesn't mean you should have files and memes ready to paste in any relevant comment thread whenever you want. That's not helpful, generally. What is helpful is knowing the relevant facts surrounding the issue; news, statistics, research that change or bear on the facts; also, the real understanding of your opponents. You should know both the prevailing and the most well-reasoned points of the opposition. This goes beyond knowing your enemy for the sake of defeating him; you must know and understand your enemy because he is a valuable human being worth hearing and being known, same as you.
Don't be afraid of being wrong, but be afraid of being wrong. It's inappropriate to live your life on the fence. I personally have a hard time with this. Since I can see the rationality on both sides of most things, I don't like to make my permanent mark on either side. Truly, it is better to be passionately wrong than to be half-committed to everything. Of course even true and honest conviction may be wrong. Your understanding of Scripture might be off. What you thought happened maybe didn't. What you thought you heard or knew or read or learned, you actually didn't hear or know or read or learn. You can't be too committed to your own understanding; have a healthy fear of your own fallibility.
Adapt. Some conversations are better with heavy doses of satire and tongue-in-cheek humor; in some situations these comments would be inappropriate and rude. Sometimes, seriousness is viewed as haughty, and using words outside of the common vernacular may well be an ugly show of pride.
Be certain about what you ought to be certain about and uncertain about things that are uncertain. Your mind will be quick to categorize things you don't actually know for sure as facts. It's the high school experiment where the teacher stages a mock-theft and the witnesses have to recount the incident. Their accounts vary widely even though they all saw the same event. Regardless of whether you took any notice of the perpetrator's clothing, you will swear he was wearing a black t-shirt and blue tennis shoes with untied laces. There are facts. There are things that are absolutely true, and we can know them for certain to be true. There are things that can be true or that are probably true. And obviously, there are things we may think are true, but they're not at all. Bias affects our understanding, what sources we trust, what truths we propagate and which ones we let die. Happens to me a lot, and it's very frustrating. It's hard to realize it in the moment, but try.
If you can put your emotions down on a shelf somewhere, do it. Be passionate. Speak when you ought. But don't go blubbering about with your heart hanging out of your shirt all the time. Love and compassion are usually necessary -- and good; whereas, indignation is not. Pick your anger up off the shelf to feel later if it's still there; in the moment put it away.
Accept disagreement and go ahead and argue. It's not wrong or rude or unpeacemakerly to have significant conversations about significant difference. Don't be too quick to "agree to disagree." In short, be real, and treat your opponent, even your enemy as a real person too.