Thanks for the responses.
I guess I ought to clarify a bit.
When I was a lad, I taught myself using a beginners book that worked as pulykamell described: chords with the left, melody - one note at a time - with the right.
So, that’s how I learned the chords. I have advanced quite a bit, though. I play almost exclusively popular music, and I’ve taught myself to do some interesting things with my chord knowledge, such as rhythm changes, chord ‘runs’, etc. But my left hand is always playing some variation of whatever chord is called for.
With my right hand, I guess I do play some chords (but I don’t really consider it as such when I’m playing). For example, if the tune calls for a B flat chord with a B flat note for the melody, I’ll play the chord with my left hand and maybe an F note and a B flat note with my right.
I guess I’m basically improvising, or faking. I’ve been doing it for so long that I can get through most pieces of popular music without much difficulty after a run-through or two. I can sound semi-impressive to those who do not know any better.
I should sign myself up for some lessons; I’m likely doing some things incorrectly.