As you might expect, the answer varies quite a bit. For me, in particular, I'm headhunted about twice a month by headhunters/recruiters who value one of two things (or both): MBB brand and data/business analytics background.
About half of them are specifically looking for ex-MBB (i.e. won't accept tier 2s).
The amount of headhunting goes up based on 1) My toggle for looking for work status and 2) My location (i.e. far fewer when based on Melbourne than when based in NYC).
Furthermore, you're absolutely able to reach out to headhunters yourself. When looking to leave BCG, I directly reached out to a few placement agencies. They weren't amazing (I ended up following a former colleague to a strategy analytics company they had been in for a year), but they were useful and laid out some great Plan Bs for me.
Hope this helps!