the keys of D or G are the safest bets.
the chords are all G major chords but the sequence doesn't Start on the I
D-D-C-D , D-C-G-D
in the Key of G would be:
V-V-IV-V, V-IV-I-V
But since it starts and ends on D it might be better to think of it as a D mixolydian based chord progression (which is a mode of G as you thought):
D D C D , D C G D
in the Key of D would be:
I-I-bVII-I , I-bVII-IV-I
If it were strictly diatonic to the key of D major, the C would have to be a C#(dim), but that would completely change the sound.
in short, I would call it a mode of G : D mixolydian.
That said, your note choices should be more dependent on the chord of the moment than what key or mode the overall progression is. I personally wouldn't think to hard about the specific mode of the piece unless explicitly called for.