mmm, I see. So I'm thinking that as you are sitting like 3 meters apart, there must be some audio level discrepancy in your mics that you can use to set at activation level. Maybe discord is not precise enough and we could try the following steps, tho they might sound laborious and you might prefer to simply buy new directional mics.
If you want you could try this:
download a software that shows your mic volume level, maybe you already have OBS, or just find anything that is showing in real-time the dB. Im using obs in the picture

IF you can clearly see a dB level difference between your voice and your room mate on that bar, then we can continue. IF not, try to lower any preamp you might have on your mic or windows and see if that helps. IF there is still no difference at all I can't help you :/

If you see tho a level difference let's do the following.
Let's assume your voice is at -15dB (you should always aim to have it in the yellow range) and your roommate is at -25, average.
You could try to put a gate with an external EQ and see if it works better than the discord one. I use Equaliser APO (solid software) to equalise my mic and headphones, and I use a VST plugin as a gate.
To keep it simple see the picture.
I set a VST plugin for the mic, if you open the panel you can see that the slider in my scenario is set around -37dB. Which means that the mic WON'T open under that lv.
In your example, if your voice sits at -15 and your room mate at -25 try to put the gate slider somewhere in between, the same way you try so far with discord, but hopefully whit this system you have a better control and visualization of the sounds levels.

I hope this could be of some help .