It has about as much atmosphere above the point where the pressure is 1ATM as we have on Earth. And no, we don't know if Earth's magnetosphere ever filters most of the incoming radiation (we don't such good measurement of the incoming radiation), what we know is that at sea-level, our atmosphere alone is enough.
AFAIK, every time we measure it better, the effectiveness of our magnetosphere decreases. But it can only stop charged particles anyway, and air is very good at stopping those.
>And no, we don't know if Earth's magnetosphere ever filters most of the incoming radiation (we don't [have] such good measurement of the incoming radiation)
Where did you hear that?
The space station (and presumably other satellites) is perfectly capable of measuring this.
The space station is in low Earth orbit, so quite low inside the earth’s magnetic field. That’s why the radiation flux shown in the first chart you linked is so low relative to Mars, even though the earth is much closer to the sun. However we have sent out multiple probes able to measure the radiation environment away from earth. There’s lots more to learn, but we already know a fair bit.
AFAIK, every time we measure it better, the effectiveness of our magnetosphere decreases. But it can only stop charged particles anyway, and air is very good at stopping those.