Who knows where it comes from? I don’t know about the industry average, but I don’t think that there would be much variety in this matter. That is: if there is something important and clearly visible in English, then there is a high chance it will get outsourced to native translators – it’s another issue that said translators don’t necessarily put in an effort, and the Japanese staff first of all don’t know enough English to realize that, and secondly even if there is someone to point out derps, production-side people don’t really have a say in the matter.
On the other hand, when it comes down to minor stuff that’s not important or just isn’t on screen long enough, it’s usually not worth spending money on a “proper” translation, and instead someone in the studio does it. It’s usually the peak of my day when people ask me to do something like that, since that means I get to contribute something more directly than scanning keyframe drawings or picking up finished material from tweeners.
But what if there’s no foreigner in the studio who you can conveniently ask about stuff like this? Well, I can only guess since for obvious reasons I don’t (can’t) get to see such a situation. Most likely then comes asking the internet and throwing in what little English their 400 points in TOEIC covers. That pretty much explains why more often than not there are ridiculously wrong pieces of English text in anime.