I know this question is marked as answered, but there is some missing information.
First, the A that you have circled is definitely A♮. There is no A♯ in the key signature, and the natural sign in the treble staff is simply a cautionary accidental. The accidental is there to make sight-reading easier, not harder, since there was an A♯ in the preceding measure. The natural signs could have been placed in parentheses or omitted entirely and the meaning would not change. Both staves have A♮ in the second measure.
Second, 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.