I've done it on solo rides, but I don't see it as competition. If I am trying to catch another rider it's really to see how fast that guy is riding. I won't, however, try to grab his wheel. I will hang back a distance and try to maintain his pace. Then I either go a different direction or we end up at a light together, in which case I'd say hi, where are you headed, and then usually let him go ahead. If he's cruising at pace slower then mine, I'll pass.
I've done it often as a runner too, but again, I don't follow close enough to make the other person feel uncomfortable.
It's not a competition if you're the only one thinking it's a compeition.