Can anyone think of a conflict or disagreement - could be political, social, moral, legal, military, etc. - where both sides are right or justified?
There are many disputes where both sides have a legitimate point. Palestine, abortion, death penalty to name a few. Depending on your perspective, however, only one side is right.
A divorce.
You could have a situation where 2 parties have changed where they are no longer compatible. Both parties disagree on the direction they would like to go as a couple, so they then both agree on a divorce.
Actually in a lot of situations they both don’t agree on a divorce: one of them wants to remain married.
Some of the times that I see both sides of an argument being right, it’s because they’re arguing the opposite sides of different arguments: pro choice vs. pro life, gun control vs. right to own guns, same sex marriage vs. freedom of religion…
Anything where the disagreement is a matter of aesthetics. Strawberry vs chocolate.
Where did I say “every divorce”? 
OP asked for examples where there was a conflict and both sides are right. A divorce is an example where those conditions could be met, not “all divorces”, but “a divorce”.
I agree. That’s what makes so many disputes so… disputed.
I think the Israel-Palestine issue qualifies. Both groups should have a secure state, unfortunately they either overlap or coincide.
Also Coke v Pepsi, Burger King v McDonalds, etc.
Today I was listening to a podcast of This American Life. It was a story about a program in Richmond, CA that gives money and support to “bad guys” in the community who want to rehabilate themselves and live a straight-laced lifestyle. During the program’s first year, the bad guys got $1000 each just for showing up on the first day.
One side says “Wow. That’s incredibly unfair. Here I am, busting my hump to be a good guy with no hand-outs or hand-holding, and these criminal elements are getting paid just to not shoot people anymore. Maybe I need to take my cues from them and kill someone!”
The opposing side says, “Yes, it is unfair. But it’s also unfair that these guys are in this position in the first place. Most of them were born into broken homes and received inadequate education. Most of them are in so deep that they can’t get out of it unless someone steps in and lends a hand. Besides, you’re either going to pay up at the front end now, or in the back end when they go the prison. Or when they kill you. At least this way we have a chance at some success stories.”
Both sides are right. But one is idealistic. The other is pragmatic.
There are lots of things where I think both sides are right. Still others where I at least understand the impulses of both sides.
Legalized prostitution. Legalized hard drugs. Forced psychiatric medication.
I tend toward the side of personal freedom, but I recognize that all of these have pretty shitty repercussions for society as a whole.
Which side is which? Until it’s been tried, it’s an experiment. You could say that it’s an idealistic experiment, but maybe not.
If it ends up being successful, and is proved to be a good investment, then it’s pragmatic.
The American Civil War.
The English Civil War.
The Thirty Years War.
The Russo-Finnish “Winter War” of 1939.
(Although one might argue that these last three are examples of wars where both sides are wrong.)
Only one side was right in the American Civil War. The CSA wasn’t it.
In the Civil War, the CSA was fighting to preserve slavery. No way was that right.
The Union wasn’t fighting to end slavery, at least not at first. They were fighting to preserve the Union, slavery or no slavery.
If you wanted to argue equivalence, the best you could do is argue that both sides were wrong, but there’s no way that the CSA was right.
To the degree they were fighting for separatism and regional self-determination, they were right. The U.S. seceded from Britain one pretty much the same principle.
The specific issue of slavery makes the CSA very, very wrong. But they did have a valid issue in terms of secession per se.
I’m glad we beat the sons of bitches…but it’s wrong to deny that self-determination was part of what they wanted, and that self-determination is a valid reason to go to war.
The U.S. has, on numerous occasions, supported secessionist/separatist factions in wars throughout the world. It is a principle that we mostly admire.
The calls for separatism and regional self-determination were about slavery. “States rights” is almost always a dog whistle for some sort of repression in this country. The one exception I can think of is marijuana laws.
The revolution against Great Britain was about taxation without representation, which is a completely different issue.
Hell anything like above where people are arguing over their opinions and who/what they like better, though whether they’re right depends on the person. I will say their debate is somewhat meaningless due to arguing over subjective stuff like these:
Heath Ledger Joker vs Jack Nicholson Joker, mlp vs mlp:fm, Superman vs Batman, Dark Knight vs Avengers, etc.
and when it comes to the internet, people are not going to easily change their minds all the time.