Polish is my native language. I don't speak any other Slavic language, but due to the similarities between the three, my Polish helps me understand the general gist of what a person means when they speak Czech or Slovak. I've tried to understand people speaking Ukrainian or Russian on occasion, but that's a bit too far. I can sort-of make out a few words here and there, if I'm lucky.
English I've learnt, but by now I consider myself proficient enough that I might as well be a native speaker (I do think I've passed the point that most learners of a language never do, where it doesn't feel like it's a foreign language that I'm speaking. I even think in English with some frequency.)
This does not have direct impact on any language skills, but I studied Linguistics at college (in an English Studies department), so that helps me with understanding the inner workings of language in general, and of English specifically. I figure I'm more comfortable with early Modern English (say, Shakespeare) than most natives are. It always surprises me a bit that Britons or Americans tend to have trouble reading Shakespeare without annotations...
I studied French for quite a few years in school at then some more at the university. I forgot more knowledge than I retained, sadly. I understand written French perfectly well as long as I have a dictionary handy to check a few words here and there. Spoken French is tougher, I struggle to understand anything more than simple conversations, but if I talked to someone who speaks slowly and patiently, I'd probably manage.
On the other hand, I wouldn't be able to speak or write in French beyond the VERY basics. Mostly due to how complex French grammar is (said the Polish guy...). I plan to re-learn the grammar, though, and expand my vocabulary, because I feel like much of the knowledge is still in my brain, buried deep, and I could speak decent French if I put a few hundred hours of work into it, to refresh my existing skills and fill in the gaps.
Since (written) Spanish is pretty similar to French, I could probably grasp the basic message of a simple text due to similarities of vocabulary, if I strained. That'd be even trickier than trying to understand Czech or Slovak, though.
I studied German for three years in High School. I can understand a few words and maybe read a few simple sentences based on mostly guesswork. Not much beyond that, though.
I'm thinking, in addition to re-learning French, about learning Korean. I even learnt the entire alphabet once, as a tentative first step (it's super-simple as writing systems go and much more logical than ANY other writing system I've ever seen, I literally learnt it in an hour or two... then promptly forgot everything due to not practicing even once). Lack of time keeps getting in the way, though.
Also, I love to pick apart the grammar and etymology of languages I don't speak (reading Wikipedia articles about the grammar of random languages is something I do surprisingly often). Also, to learn songs in languages I don't know a single word of. I'm weird that way.
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EDIT: Tlaloc,
I attribute this to the fact that I don't really remember learning grammar at school in English, and the grammar is generally so simple that you can kind of pick it up without making any conscious effort. Not so in other languages, I've found.
It's not actually due to the simplicity of the grammar. If English is your native language, that's what the difference is between learning a language and acquiring it, according to some theories in linguistics. Languages acquired early in life are not picked up as a skill, but an almost innate ability that our brains are hard-wired to pick up. It changes later in life, that's why you have to pick up theory and rules and learn exceptions by heart in foreign languages. That's also the reason why uneducated people can always speak their own language grammatically, correctly and with no effort. Their brains don't need to "learn" much, when the person is young enough that the language "settings" of the brain have not... solidified yet to match their native language (or languages).
This is related to the theory of Universal Grammar, which Wikipedia explains pretty well if you're interested:
https://en.wikipedia.org/wiki/Universal_grammar