But if you then feel entitled to treat that person like dirt, you know what that shows? You're indecent.
That's what we need to correct right now more than ever. You don't have to change your denomination of Judaism or your political affiliation in order to be a more decent Jew. You only need to engage in good behavior toward all the people in your life (this could obviously extend to non-Jews as well). It's not the end of the world if we disagree with one another, but if we therefore don't regard each other as equal human beings, it is the end of the world. In order for society to function, individuals have to maintain a certain decorum around those with whom they do not get along.
We don't necessarily have to go out of our way to hang around people we don't like. In fact, depending on the situation, it could exacerbate the problem. We just have to treat those people as decently as possible when we are around them (e.g. on the street, at a wedding, bar mitzvah, etc.). Prepare yourself for these situations and you will react much better. For instance, see to it that you acknowledge their existence and say hello - or if they say hello first, politely respond. If you happen to have an unresolved problem with them, you don't have to pretend like everything has been magically fixed - but if the issue comes up in conversation, simply state your case in a kind manner. Be clear but courteous.
Another possible situation that we may encounter is when our personal enemies need help. For example, if you are walking out of the grocery store and see two people who dropped their groceries - one person you get along with and the other person you do not - our Sages (Bava Metzia 32b) derive from Exodus 23:5 to help the person you don't like first. Notice how the Torah understands human nature - certain people will inevitably have trouble getting along with others. However, the Torah also teaches us how to deal with it - we must demonstrate that God-based goodness means more to us than personal vendettas.
The bottom line is this: while it may be natural to harbor less than pleasant feelings toward fellow Jews every once in a while, we cannot allow ourselves to act upon those feelings. Just because we personally don't get along with someone does not mean that they deserve to be treated poorly - or worse yet, embarrassed and humiliated. As long as we do our best to follow the Torah's basic code of interpersonal conduct, we'll be alright. God doesn't expect us to be perfect angels, but He does expect us to be good people.