Yep, with the desktop versions of Signal, Matrix (Element) and Steam chat.
I'm yet to try out gamemode, which may help. But I typically close Signal and Element when I'm gaming, so it's usually not an issue for me.
Edit: I'm on EndeavourOS, KDE, i7, 16GB, Nvidia 2060.
I'm assuming work and personal?
I've done the same for years, but am thinking of making use of the second SIM slot in my personal phone and using profiles to separate everything. Many phones can also use an eSIM, so that's also an option for multiple numbers.