Note: This answer refers to the chord symbols ("Gm", "C7", "F", etc) found on top of the song staff. Colonel Panic's question was actually not about this, but about the notated chords in the upper piano treble staff. I'm leaving this here since it might help someone else.
The chord notations/symbols are merely provided as support, or for letting you make your own accompaniment.
So, either you play the two piano staves as they are written ignoring the chord notations (other than as a possible support in reading the written music), or, if you wish, you can ignore the written notes and make up your own accompaniment from the chord notations.
(The chords are already in the written piano staves, so -- no don't try to add them to the written music.)
The chord notations can also be useful for other accompaniment musicians, such as a guitarist, in playing along with you or as the sole accompanist, or for, say, a saxophonist providing backgrounds and embellishments to the singer and your playing.