Since teaching English as a second language both in South Korea and now in Japan, I can safely say that it has affected my writing, and perhaps my speaking ability.
First, I'll start with the good stuff. When I have students ask "Why?" about parts of English grammar and things that are so natural to me as a native speaker and writer of English, I have to stop and actually think about why something works that way. If I were speaking to another native speaker, I'd give some excuse like, "It just sounds better that way."
But that doesn't work in Korea or Japan. For one, neither Korean nor Japanese has plurals. So when a Japanese student says, "I like cat," I'm pretty sure they mean, "I like cats." But if a Korean student made the same mistake, I wouldn't be as sure about their true meaning. That's mostly because saying "I like cat" is like saying "I like to eat cats," which is a perfectly acceptable food in Korea.
Now for the bad stuff. Since coming here to teach English, I'll make sentences like, "I started working on this story longer ago than I started working on Kitsu's story," and I'll stare at the sentence thinking that something's wrong with the phrase "longer ago," but I have no idea if that's a viable English sentence or not. Though I'd like to keep the "when in doubt" thought going, and hopefully assume I'll be smart enough to get rid of this questionable phrases in my writing. (Unless of course I'm writing a first person story about a girl who's teaching ESL abroad and want to keep her messed up English "authentic," but I don't see that happening anything soon.)
I'll also look at a work that my subconscious knows is spelled correctly, but for the life of me, I think it's misspelled somehow. Which can be a pain when writing everything out on paper where I don't have a spellchecker to tell me if I spelled the word wrong or not. But, this one's really only a minor thing since I'm pretty bad at spelling to begin with.
And the last one is that I never thought about English as being "fresh" or I suppose "stale" until I came to Japan. I had one of the other teachers at my school who's been in Japan longer than I have ask me for help with one of the "Why?" questions of English grammar because my English was "fresher" than his. (Actually, the "fresher" is mine. He said, "Hey, your English's fresh.")