I'm English and most of the people I know get all of the examples you mentioned wrong every time, especially 'their', 'there' and 'they're'.
Foreign speakers often have a more comprehensive knowledge of grammar because they learned 'by the book'.
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Well, I learned before I could read, so it's not really by the book. But I do get what you're saying. Grammar knowledge is only part of it though. You need to familiarise yourself with the flow and feel of each language.