Whenever we discuss the key of a piece, we usually describe it as either a major or minor key. And then, based on the intervals of the major or minor scale and the root note we define, we can construct the sharps and flats of that key.
When we play a mode of a major or minor scale, for example the mixolydian mode of the major scale, what key are we in?
Consider the mixolydian mode of the C major scale. C D E F G A Bb C
If we played a song in this scale, the song would have the same key signature as the F minor scale, (though it would have C as the tonic).
So what key are we in if we play C mixolydian? Are we now in F minor?
If I were to rephrase this question more broadly, it would be this:
If keys of songs defined in terms of the major and minor scales, how do we define our key when we play scales other than the major and minor scales?