Hook the output to a speaker.
May I have some more details, please.You could run it through a high-order low-pass filter to remove the harmonics that generate the harsh sound, leaving only the fundamental sine wave.

Thanks for your input.Below is the simulation of a 1kHz squarewave into a 3-pole 1kHz low-pass active filter.
The filter removes most of the squarewave high frequency components, leaving the fundamental sinewave with reasonably low distortion V(out).
For comparison the output is also shown from a 1-pole passive 1kHz RC LP filter V(out2). As you can see it looks more like a rolled off squarewave rather than a sinewave.
Note that the filter is basically for a single frequency and the output amplitude will be reduced as the squarewave frequency is increased.
If the frequency is reduced the distortion will increase.
If you don't mind some distortion in the output then you can use a simpler filter.
Here is the website I used to generate the filter component values.
View attachment 79717

Okay, then which of the techniques mentioned in the link should I start with to have a harmonic-free final signal? And again, do I need to introduce two sine-wave generators for each frequency and then mix them?Yes, you would need a separate filter for each frequency you generate with the filter corner frequency equal to the square-wave frequency.
Edit: If you need a sinewave then generating the sinewave directly, as bertus suggested, might be a better choice.
Thanks for that valuable stuff.something else to read
http://www.ti.com/lit/wp/snoa839/snoa839.pdf
you need to scroll down some in the next one to find circuit schematics
https://www.calvin.edu/~pribeiro/courses/engr332/Handouts/oscillators.pdf
Of course, I do.perhaps abuhafss wants to retain a bit of the siren sound
Do you mean the outputs of each 555 or the output labelled OUT-MIX?I'd suggest hooking a 100k pot in series with the output, and a 0.01uF cap on the far side of that.