What you are asking about doesn't really have to do with modes or accidentals. Essentially what you are talking about is the difference between modern equal temperament on the one hand, and just intonation on the other.
Up until the late 1800s, musical instruments could play the traditional Greek modes and scales based on pure intervals, but a given instrument could only play in tune in a few keys, certainly not all 12 keys. Since the late 1800s, a new system in tuning instruments was developed: that system is called equal temperament.
With modern equal temperament, our instruments are calibrated to play 12 notes in an octave, all of which are equally spaced apart by 100 cents. This system is only about 125 years old.
Before that, people used various kinds of compromises on just intonation, where intervals are pure. In a just-intonation C-major scale, there are half-steps and whole steps, and we call the pitches by the same names that are used in equal temperament, but the various half-step intervals all have a slightly different size, as do the whole-step intervals.
The music written by Bach or Mozart or any of those composers, and certainly all of the ones that came before, was not composed to be played in equal temperament like we usually do today.
This is a deep and complex subject that requires a lot of mathematics to explain, and I don't choose to go into that here. You can find plenty of references on just intonation online.
Here is one observation for you to think about: On a modern piano or guitar, which is tuned to equal temperament, all the major third intervals are considerably sharp compared to a pure, just-intoned major third. The equal-temperament system was developed to permit an instrument like a piano or guitar to play all the modes in all 12 keys and have them reasonably in-tune, but not perfect. Before about 125 years ago, a piano would be tuned to a mean-tone system, a modification of just intonation, that enabled certain keys and certain intervals to sound really in tune, and certain other keys would sound really out-of-tune.
Today, when singers in an acapella choir sing together, they can make pure major-third intervals (and other intervals as well), and so can a string quartet, where the instruments have no frets. But if singers or string instruments play along with a piano or guitar, they can no longer make those pure intervals because they clash very slightly with the equal-tempered pitches on the piano or guitar. So, really without even thinking about it, singers and string players adjust their intonation away from just intonation and toward equal-temperament, depending on the situation.
None of this distinction between just intonation, or meantone intonation, or equal temperament, has any impact on how the music is notated. We use the same 12 pitches and the same sharps, flats and naturals (although G-sharp and A-flat may be distinct pitches which are tuned slightly differently, for example). The difference is how it is performed and on which instruments and how they are tuned.
Again, this is a deep subject and I can't thoroughly explain it without writing a book on it, but there are plenty of books already written on the subject and you can go looking for them if you would like to.
Update: See my post from December 23, 2012 where I present a chart showing the differences between equal temperament and just intonation, measured in cents. (A cent is 1/100th of a modern equal-tempered half-step interval.)