Let's say you were writing something in B-flat major. And at the beginning of one measure, you have a natural on an E-flat in the left hand. Would you also  write a natural on an E if there were one in the right hand in the same measure? Or do you assume that the natural applies to every E in that measure, not just the E that is one the same line/space as the first E?
ah, that is a good question. i'm sure everyone will chime in saying it's one way or the other, but from what i've seen, there is no clear answer. that's why it's good to use 'courtesy accidentals' which is when you put the accidental (that's the sharp/flat/natural symbol) in parentheses