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For example, when reading piano music on a grand staff, I know that if there is an A# in the treble clef staff (where it's normally natural), it changes all of the A's in the rest of that measure to A#.
Do all of the As in the bass clef staff stay natural unless they also have a # symbol next to them, or are they all affected by the # in the treble clef?
The answer to the question in the title of your post is, "it depends." As a general rule for Western notation as it is practiced in the United States, the answer would be "no"; not only is each staff independent with respect to accidentals, each octave is independent with respect to accidentals. However, when an accidental is present in one staff but missing from another, and the accidental is not merely cautionary as in your example, a misprint is likely unless you know that such a "clash" is within the style of the piece. The same applies to an accidental missing from one octave of a staff. Within a staff, however, in some French publications, an accidental applied to one octave is intended to apply to all octaves. This phenomenon appears with some frequency in the "classical" saxophone literature. I would expect different staves still to be independent in this case, but again, misprints are possible.
tpburch is correct about the grand piano staff, and that would also apply to other grand staff instruments, such as marimba.
However when in the case of clefs being for different performers, such as choral scores, orchestral scores, brass quintets, etc., each clef is treated on an individual basis.
Strictly speaking, the accidental ONLY applies to the corresponding pitch in the same octave, therefore you are a little incorrect when you say:
I know that if there is an A# in the treble clef staff (where it's normally natural), it changes all of the A's in the rest of that measure to A#.
I think it's dreadfully unclear if there is no cautionary accidental for the same note in a different octave, and the composer/arranger does not intend that note to be affected by the accidental in another octave, but any cautionary is for clarity only, it is not required according to "the rules":
“In modern usage a sign is valid for the note that it precedes (but not for the same note in octaves above or below) throughout the rest of the bar, unless expressly contradicted by another sign.”
(The Oxford Companion to Music, my emphasis)
Furthermore and to answer your question: accidentals do not carry through to different staves in a grand staff, so the A's of your example would remain natural.
A common misconception is that accidentals affect future notes in the same measure across octaves. This is simply not true. Accidentals only affect the note in the octave in which they occur. So, if A4 has a sharp accidental and there is an A3 later in the same measure, the sharp doesn't carry over. This happens often in jazz music, as well as in neoclassical music like Prokofiev.
An accidental on a note should affect all of the same note (regardless of octave) for the rest of the measure. This is the case for piano, where one performer is playing multiple clefs.
However, for an orchestral score, I would assume that each line is autonomous, whether in the same clef as another instrument or not.
As far as I am aware, the common practice is to write the accidental in each clef for the sake of clarity, even when not specifically required.
You may need to distinguish between accidentals and sharps and flats that belong to the key. If the key is D sharp then that is the key for all the clefs in a specific bar. There may be some reasons that notes are raised or lowered that fall outside issues regarding key.
There may be some chromatic passing notes that may very easily have sharps or flats in them that do not signify a change of key. They may also be specific to a clef and not necessarily happen on all the clefs / voices of a piece.
