I'm going to try to keep this as bland as possible...
I grew up Catholic in an area of the country that due to Catholicism and one industry, usually voted for one party.
I've now been an Orthodox Christian for almost seven years (plus one year of attending before I converted). For five years, I was in a parish that had a lot of converts from Evangelical Protestantism, so you can guess their political leanings. I tried, but I did not fit with their thinking. The priest even had that leaning - which you knew from personal conversations, plus if you read between the lines when he preached, although he specifically kept politics by name out of the pulpit, but stances on certain social issues were repeatedly brought up.
I've now been in another Orthodox parish for more than two years. Much more mixed group. Our priest will bring up certain social issues we are against, but he doesn't hit us over the head with them like my previous priest did. I only know my current priest's political stance from personal conversation and some things he's written on an email list for our faith tradition. Current parish is much more mixed in makeup of members (who votes for what party). My priest did recently insert some petitions for the end to the Middle East wars in a litany that allows for such insertions, but since we already pray for peace anyway, this wasn't a big deal. Since our priest has set a good example, we have no fear/hate mongering discussion at coffee hour. Political discussions are kept pretty quiet. If people are of the fear mongering mentality, I don't know about it. My priest knows I have the same stance as he does, and we'll occasionally have related discussions but they are on email or in a group of similarly-minded parishioners.
I am very much bothered when people of my faith go overboard with the fear mongering you see among some Christians nowadays. My FB feed is filled with all sorts of extremely negative articles. These are well-educated, sensible people, but some of them seem to have just gotten rather, well, extreme.
I absolutely refuse to talk politics at all with friends from my old parish. We have lots of other stuff to talk about though. One or two suspect I have a different thinking from them now, but they don't pursue it. I also refuse to put anything political on my FB page. The closest I might come is something about the environment, but my friends are at least polite on my page and don't make snarky comments. I will also post interesting articles that show faiths cooperating and not arguing, such as a recent article from the NY Times about traditional people of different faiths living together in NYC. My favorite part of that was the two Muslim women and the one Catholic woman, all very observant, who were roommates. They got along great and were all bridesmaids when one of the Muslim women got married. I thought that was very cool!
My way of speaking out is to NOT say anything on my friends' extreme articles posted on FB. Since I comment on everything else, but not those, I think they've gotten the message. I'm continually astonished at the articles they dig up. I'm also disappointed that they continually drag our faith into this ugly stuff. I pray for them a lot. I'm much more into actions than words, so this is a way that works for me.
I'm a member of a peace-related organization for my faith tradition, so I have opportunities for positive discussions on our email list and among several other members locally. We do what we can to promote peace. I'm a "fan" of this group's page on FB and so people might see that, but no one has made any comments about it.
Personally, if I'm at the grocery store or other shop and a Muslim woman wearing hijab is the cashier, I will make sure to pick her register, smile, and be friendly. My cross on a chain is visible, and I want her to know that not all people of a certain faith are hostile to her.
I've come to the opinion that observant (in a traditional way) members of all faiths have a lot of good to say to each other, while still believing that yes, there are differences in belief.
I also think one's media choices play into all of this. I listen to NPR, read The NY Times online (as well as our local paper), and listen to a local AM station that has all local programming and none of the syndicated stuff that is sooooo extreme. I'm TV-free and have been for some years.
I've been thinking about this for a while, and I'm glad you brought it up. I hope the conversation can continue in a good way.