All they have to do is just respect each other's rights to life, liberty, and property. Imagine if the Republican and Democratic parties were each governments. Each citizen picks whichever one they want. If you commit a crime against another citizen of your party, your party handles it in the way they deem appropriate. If they commit a crime against a citizen of the other party, it'd be handled by collaboration between governments: extradition, etc. So, if you are Republican and murder a Democrat and the whole world witnessed it on television, the Republicans probably hand you over to the Democrats to face justice. But if you are Democrat and have an abortion, the Democrats don't hand you over to the Republicans, because while the Republicans have outlawed abortion, the Democrats have not.
Now generalize it: instead of two parties/governments, allow any arbitrary number of them. Allow people to create new ones as they see fit.
How is that different from what we have right now on an international level with the difference being only that your parties/governments (i.e. states) are distributed across the globe? But in today's world, where you can fly from any inhabited area to any other within several hours, even this limitation is actually losing its significance...