How to make scatter plots with semitransparent points in Gnuplot?

You can use the alpha channel of argb along with lc rgb variable:

set samp 2000
randn(x) = invnorm(rand(x))
pl [-3:3][-3:3] '+' us (randn(0)):(randn(0)):(0xBB00AAFF) lc rgb variable pt 7 ps 2

.

This leaves some egde around each circle, probably an opacity effect from a circle plus a filled circle on top of it. Unfortunately, there is no edgecolor option as in matplotlib to control this. Replacing filled circles pt 7 with open circles but thick linewidth pt 6 lw 6 can mitigate this a bit

pl [-3:3][-3:3] '+' us (randn(0)):(randn(0)):(0xBB00AAFF) lc rgb variable pt 6 lw 6

.

One can also emulate a variable transparency with lc rgb variable

set samp 50
pl '+' us 1:1:(0x00AAFF+(int(0xFF*$1/10.)<<24)) pt 7 lc rgb variable ps 3

where int(0xFF*$1/10.) maps the input from 0..10 into 0..255 and <<24 shifts it into the alpha channel.

Note that in your plot only the dense regions seem to have a transparency effect, but not the scatter points in the background.

Try this:

set style fill  transparent solid 0.35 noborder
set style circle radius 0.02
plot 'test' u 1:2 with circles lc rgb "blue", \
     '' u 1:2 every 100 w circles lc rgb "red" fs solid 1.0 border lt -1

which outputs

As you can see, you can specify for each data set whether to use transparency and which color to use.
If your data consist of two values (x and y position of the circle) you have to specify the circle's radius via set style circle .... If your data has three rows, you can use the third value to specify the circle's radius for each data point.
You can play with the transparency level, which ranges from 0 (full transparency) to 1 (no transparency).