I used the code here and compiled in my computer, with the result as seen in the picture (this was compiled using PDFLaTeX -> Asy -> PDFLaTeX; neither XeLaTeX, LuaLaTeX nor dvips -> ps2pdf solve this). For comparison I also did it using TikZ and transparency works fine. Here's the MWE:
\documentclass[12pt]{article}
\usepackage[paperwidth=10cm,paperheight=6cm]{geometry}
\usepackage{tikz}
%\usepackage{asymptote}
\usepackage{asypictureB}
\pagestyle{empty}
\setlength{\parindent}{0pt}
\begin{document}
%
\begin{asypicture}{name=1}
%\begin{asy}
unitsize(1cm);
fill(shift(1.5dir(120))*unitcircle,green+opacity(0.75));
fill(shift(1.5dir(60))*unitcircle,red+opacity(0.75));
fill(unitcircle,blue+opacity(0.75));
label("Asymptote",shift(1.5dir(270))*(0,0));
%\end{asy}
\end{asypicture}
\qquad
\begin{tikzpicture}[opacity=0.75]
\fill[shift={(120:1.5)},green] (0,0) circle (1cm);
\fill[shift={(60:1.5)},red] (0,0) circle (1cm);
\fill[blue] (0,0) circle (1cm);
\node at (270:1.5) {TikZ};
\end{tikzpicture}
\end{document}
I tried using the asymptote and asypictureB packages. I had a somewhat similar issue with PSTricks; the user Akira Kakuto pointed out that this was due to a bug with both Ghostscript and xdvipdfmx (versions 9.52 and 20190824, respectively; as packaged in Debian's TeXLive 2019). Is this related to that bug? I found this report, so apparently the GS hypothesis stands. Is there any way of properly diagnosing this (that is, to be sure it is a bug with GS)? Or to solve it? Thanks!

asymptotepackage (v1.33) and compiling withpdflatex,asy -nosafe yourfile-1.asy,pdflatexagain. Note thenosafe, as also mentioned in the Github thread. My versions are somewhat older, Asymptote version 2.41, Ghostscript version 9.26, all installed using Debian packages for Ubuntu 18.04. Maybe you could try running justasy -nosafeon the example.asyfile, see if the issue is there or somewhere in the LaTeX chain? – Marijn Apr 26 '20 at 20:29-nosafeoption, I regret to say. – mathbekunkus Apr 26 '20 at 21:41%\begin{asy}and%\end{asy}inside theasypictureenvironment, it will cause compiler errors when building the asy code. – ollydbg23 Apr 26 '20 at 22:39asycode compiles just fine with the current version (2.65) andgs-9.26. Also, you have%\begin{asy}in the asy code, and%does not start a comment in theAsymptotelanguage. – g.kov Apr 27 '20 at 12:31