The effect of rivals (real or imagined)
In the manga, Hikaru No Go, the rivalry between Hikaru and Akira is symbiotic. Akira's concerns for Hikaru's potential strength drive Akira onward to keep his lead. Hikaru's aspirations to surpass Akira make Hikaru work harder. In the process, both become stronger Go players.
I had a good number of "rivals" back when I started playing Go on KGS in 2005. Most were just other players of the same rank who had started playing the game about the same time that I did. I might have played a game or two with them but we really didn't know each other.
I guess you could instead describe them as randomly-selected benchmarks rather than rivals; but they served the same purpose.
Those first months were when I had the strongest desire to advance. I would often peek at the rank charts and game tables of my so-called rivals to see how their progress was going. I would then try to catch-up if I was being left behind; or experience a subtle glee if I managed to pull ahead.
However, many of these players disappeared quickly. The great majority were flash-in-the-pan beginners and they quit the game after hitting a wall near 22-kyu (on the old CGoban2 ranking system) or sooner. A small number were notably talented and quickly left me in the dust. Two, whose life profiles are similar to mine, needed to deprioritize Go and now play infrequently.
These days, I find myself just aspiring to play my next game better than my last (by thinking even just a little deeper). Obviously, as an amateur, I don't have a burning need to improve as quickly as possible. On top of that, I've probably come to accept that all folks have different life circumstances and that makes it hard to really compare one's progress to another's.
I still sometimes benchmark my advancement against other players that I've known for a while; a couple of these being Typhoon and aussiemate. I compare their progress to mine just to check if I'm "falling too far behind the pack". This gives a little impetus for me to improve; a gentle tug that helps ward-off complacency with my current quality of play.
I really hope that I don't develop some intense rivalry in Go. I'm not sure if I'd be able to handle it. I have enough competition to deal with at work.