Sorry Doc, but yeah they do. A song written in G mixolydian is written in the key signature of G major, so you will see a flat on the seventh tone (F). It's even more apparently when a scale is used over a chord. What I mean is, say your playing a tune in C major and you come to an F major chord. Instead of using the F lydian mode over the F chord, some may choose to use the F major scale instead. Causing an accidental if Bb is played. There's also something called modal mixture, which is using chord from one mode (major or minor) and imply them into a major or minor key. When this is applied to chords (chords from a mode to a key), these chords are reffered to as borrowed chords.

An example would be seeing the bVI chord in a major key. In C major, you would be borrowing the bVI chord from the C Aeolian mode (natural minor scale).