I'm going to ramble again...
Delixe: Atari ST? Well the Atari fans certainly saw Amiga fans as rivals. Amiga fans just laughed..
I think it depends when you got your Amiga (or Atari ST). When A500 was still quite new, it did seem at least to this Amiga owner that for some odd reason Amiga got much less attention than Atari ST at least from some big companies. (For some reason at least inside Europe, Atari ST seemed to get much more attention from e.g. French game companies, while Amiga seemed a bit more popular among German developers).
For example, I seem to remember how Gauntlet (or was is Gauntlet 2) first came out only for Atari ST, with no Amiga version in sight. I was devastated, but luckily Rainbow Arts released "Garrison" which was a very good Gauntlet clone. At some point the official Gauntlet games were ported to Amiga, though.
Also at some point it seemed most Amiga games were just direct ports from Atari ST, ie. they used only 16 simultaneous colors, not 32 what Amiga would have been capable of. Not to mention games like "Bionic Commando" where the Amiga version lacked smooth scrolling apparently only because Atari ST couldn't do it passably either. So, Atari ST seemed to hold back the Amiga game development for some quite time, because most games were made for both platforms. Amiga-only games seemed rather rare at least to me.
And to add salt to the injury, it seemed the Amiga ST ports of early Amiga crown jewels, ie. Cinemaware games, were much enhanced on the ST. I've always wanted to see the ST version of e.g. "Sinbad and the Throne of the Falcon", as at least in photos it seemed quite a different game than the original Amiga version. Just look how different the two versions look, heck even the C=64 version looks better than the Amiga!
http://www.mobygames.com/game/sinbad-and-the-throne-of-the-falcon/screenshots But I guess by 1989 or at least 1990 it was quite apparent that Amiga was a superior gaming machine, but at that point PC started to gain momentum against both thanks to VGA graphics and Adlib/SB/Roland sound cards...