In western movies, it is a common trope that the person to draw first usually loses in a gunfight. Is there any truth to this?
Experts draw and shoot pretty darn quickly (link).
There's a "gun-slinger's paradox", but apparently the effect of that is not very significant.
If the assertion is true I'm going to guess it's because the person who draws first is relatively inexperienced/unpracticed/inexpert.
Specifically, an expert can react, draw, and shoot within 300 msec (see link above): so unless their opponent can initiate, draw and shoot in less than 300 msec then they'll 'lose'.
In another fictional reference relating to the time period in question - namely the movie "Unforgiven", how fast you are has little relation to who "wins". It better boils down to the character of the person doing the shooting. Namely, whether that person is capable of killing another human being.
This, on the other hand, is quite well documented. During WWII, Brigadier General S.L.A Marshall conducted several surveys of riflemen who had seen combat and found that only about 15-20% would take any part with their weapons. In fact, most would prefer to put themselves into harms way rather than kill enemy soldiers. His work has been disputed, but tests involving recreations of historical battles using non-lethal laser weapons, and comparing them to historical casualty rates (referenced here) showed that these numbers were fairly accurate.
Another take on the Hollywood gunfight-in-the-street is the reason for the standoff in the first place: namely, that he who goes for his weapon first is the aggressor, and the one who shoots last is acting in self-defence. As such, with witnesses present, the former would likely be convicted of murder, and the latter would not. Which is why in old movies, the one who draws last is always the one who wins. It makes the conclusion less ambiguous and takes less time in the movie.