If we want to play a song on a keyboard, the right hand plays the melody and the left hand plays the chords. What is the case if I want to sing the song? Should one hand play the chords? Which hand should this be, and what about the other hand?
Consider the following two measures from the piano transcription of the song "You'll Be Back" from Hamilton:
In case it's not clear, the top line of music is the vocals, the middle line is the right hand piano part, and the bottom line of music is the left hand piano part.
This shows a very common style of arrangement. The singer sings the vocal melody, the right hand plays chords and the left hand plays the bass line.
Sometimes the right hand part is transcribed to double the vocal melody. That means the right hand is playing the melody and the chords at the same time and the singer would sing the exact same notes as the melody notes that the right hand is playing (the left hand usually still plays the bass line in these cases). Those transcriptions are usually meant to make things easier for the singer by giving them a melody to follow. If you listen closely to professional singers and pianists, most of the time the vocal melody is completely separate from the piano part.
First, get rid of the idea that LH chords, RH melody is 'normal'. If solo keyboard, who's doing the bass line? If in a band, your role may be to play melody, play a counter-melody or 'fill-ins' or play chords. Probably just one of those.
Back to your question. Maybe it's the sort of song that could be effectively accompanied by simple guitar strumming. Fine, emulate this on the keyboard. Very often a strong bass line would best help the song. Try this in the LH with sustained or rhythmic chords in the RH. Maybe doubling the melody is appropriate (but probably not ALL the time). Or lots of other possibilities.
Think 'support'. If in doubt, play less.
Very dependent on the song. Some benefit from a bass line with l.h. - maybe a walking bass - some from l.h. chords, some from single bass notes l.h. and chords in r.h., some from partial chords with both hands, some with a counter melody in r.h.
In fact, think of a combination, and there'll be a song in which it will ft. Or think of a song, and decide which combination fits best.But don't do the same idea in each and every song!
Assuming that you don't read sheet music, and are learning to play by ear? It depends upon the song. Many singers play chord figures with the left hand, and arpeggios with the right, during the interludes between vocals.
Paul McCartney tended to play chords with both hands, during his singing of the verses and the chorus, and subtly alternate the timing on the down-beat.
I have found that I can most easily play while singing, but usually while gently playing chords with the right hand, and striking just the bass note for each chord with the left.
Each performer develops their own style, in the end.
If you're just talking about being the piano man, then all you need is some chords and maybe a little countermelody here and there.
But if you take a look at classical piano + voice pieces (which granted can be difficult to perform solo), you'll see a full chord and melodic structure in the piano part which complements, rather than duplicating, the singer's melody.
Then again, there's no strict rule that the melody can't dive into the bass here and there with some "twinkle" notes or chords in the treble range.
If you are singing, that is the melody, so you would play harmony on the piano. Which hand plays which part is totally up to the arrangement.
One hand should at a minimum keep the rhythm. Depending on the vocal range of the singer, a counter melody, baseline (usually left hand), chordal harmony, etc. will all leave space for the singer's voice as well as other potential instruments. You could also add a melodic harmony, playing octaves, 4ths, 5ths, or some other part right along with the vocals.
When the phrasing of the singer is paused after a line, the piano parts can alter quite a bit and temporarily run into the range of the singer for some sort of riff or fill - then back out again before the singing resumes.
Of course, you can throw every bit of structure out the window whenever appropriate for the desired effect of the song! If the singer is amplified, you can collide a bit more, as long as the melody, or lead, can still be heard.