sunshinecorp: Or rather, only Han shot.
rtcvb32: To add my only two serious cents to the discussion, what does the original book say, since that would be the source material, which from what I understand is close to 1:1 to the movie. If Lucas says Han didn't shoot first, then he didn't shoot first; Instead mistakes were made during making the original film.
But what do I know...
Just to answer your query, here's what the original 1977
book, not screenplay, has in it:
*****
Something which might have been a laugh came from the creature's translator. "They'd hardly notice. Get up, Solo. I've been looking forward to this for a long time. You've embarrassed me in front of Jabba with your pious excuses for the last time."
"I think you're right."
Light and noise filled the little corner of the cantina, and when it had faded, all that remained of the unctuous alien was a smoking, slimy spot on the stone floor.
Solo brought his hand and the smoking weapon it held out from beneath the table, drawing bemused stares from several of the cantina's patrons and clucking sounds from its more knowledgeable ones. They had known the creature had committed its fatal mistake in allowing Solo the chance to get his hands under cover.
*****