In deciding whether an accidental should be printed, one should generally expect that people reading music will be blind to staffs that don't contain their part(s), and may be blind to music which they are not performing even when it shares a staff with music they are; one should add cautionary accidentals if necessary to ensure that such possible blindness will not affect the pitches to be performed.
In your example, a keyboard piece in C major has two treble clefs, and the left hand (lower staff) plays a mid-staff B♭ near the start of a measure, then if the right hand plays that same staff note later in the measure it should be explicitly marked with either ♭ or (♮). Indeed, such a marking should be used in cases where the staffs used different clefs (e.g. if the left hand, notated with bass-clef, played the F♯ above middle C, and then the right hand, notated with treble clef, played its first-space F.
Note also that if e.g. in C major one performer performs a B♭ in the middle of treble clef, and another performer simultaneously performs a B♮ in the middle of treble clef, it may be helpful to mark the latter note with a cautionary accidental (♮). Such a marking may not be necessary from a performance standard, but in its absence the dissonance might make performers wonder if the "B♮" might be actually be a misprinted B♭. In cases where confusion is unlikely, no accidental is required, but when confusion would otherwise be likely, a cautionary accidental can help.
Returning to your example, the B♭ shouldn't apply to the right hand, but I think that notationally it would have been appropriate to mark the right hand B with a (♮).