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That said, look at Latin:

• dexter - right side, but also: favourable, fitting, proper (cf Spanish diestro)
• sinister - left side, but also: adverse, hostile, bad (cf Spanish siniestro)

The "privileges" that you see in derecho and right are an extension of what Latin already associated with dexter - things that are proper to do or to get. For example if I got a right to freedom, that means that it's fitting for me to get freedom, you know?

Based on that odds are that Spanish simply inherited the association, and kept it as such even after borrowing izquierdo from Basque and shifting directus→derecho from "straight" to "right". While English borrowed it, either from Latin or some Gallo-Romance language.

And overall you'll see a fair bit of that in the Western European languages, regardless of phylogenetic association, since languages clustered near each other (i.e. a Sprachbund) will often borrow concepts and associations from either each other or from a common source.

Also, note that right "as side" and "as privilege" are not homonyms. Those aren't different words from different sources, it's the same word with two different meanings, this is called polysemy. The same applies to derecho.

Wait ‘till OP finds out that Sinister was originally the word for “left”

I don't recall all of the fun fine details of it, but waaaaaaaay back when, there was a math nerd who was also just as much of a religious nut. He's the main idea behind why graphs work the way they do. Up and to the right is good, where as down and the left is bad. The direction right became synonymous with godliness and the left direction was for evil, just like up and down. In a weird way, math is hella religious

I think the short answer is that we all speak "Latin" and the various empires, especially Roman, spread cultural concepts across Europe that would be durable for some degree for centuries.

Counterpoint, there are also plenty of false cognates between any given European languages.

There is also convergence, as another commentor pointed out. If something is invented in the US in 1970, or introduced to the US in 1970, there's a good chance people aren't going to give it a name in their language: they will just call it what it was called to them.

I don't think "loan words" explain your example, but you can imagine that if what is now Spain and France had the same rulers, they'd develop similar legal culture later on, and then if France conquered what is now England, that eventually English would inehrit specific quirks common to Spanish

I have done professional and amateur translation work in a couple of different languages. In addition to what others say, it seems like many languages go through a type of convergent evolution. For people who live in completely different cultures, with completely different histories and values, it is amazing to see how many words expressing abstract concepts can be exact translations in almost every way. It is just my intuition, but I believe translators play a key in a language's evolution.

You might find this video interesting:

https://www.youtube.com/watch?v=P1houMl12OU