What is an example song, and the particular sheet music you've used?
Most pop song sheet music is arrangements for piano and not real piano parts. Those particular arrangements hold no special place as the only, or correct way to play "the song."
You can do your own "arrangement" of a song, either rehearsed or spontaneously.
With most songs the principle thing is to know the melody and chords.
A lot of sheet music arrangements put chord symbols above the musical staff making it easy to know the chords. (Sometimes those symbols are needlessly complex, technically trying to account for non-chord tones in the melody, but let's skip that detail for now.) If chord symbols are not given, you should analyze the music to figure out the chord changes.
When the song is reduced to a melody with chord symbols, you essential have a lead sheet. From the lead sheet you then play the song, not literally as notated, because the lead sheet doesn't provide full notation, but according to different styles of accompaniment and harmonization. A typical approach is to play the melody literally, or maybe with some embellishments or a bit of doubled notes, in the right hand, while the left hand accompanies the melody playing rhythmic patterns based on the chord changes.
The simplest accompaniment is just "block" chords in the left hand, in close position, on beat one of each bar. Other accompaniments might be waltz style with bass notes in the low range on beat one, chords in the mid range on beats after one. You could also arpeggiate the chords in the left hand. If you arpeggiate with open voiced chords in the piano low to mid range, it creates a very resonant sound.
There are lots of ways to make your own song accompaniment, harmonization, and dressing up the melody. Try looking up playing from a lead sheet.