Does Hawking radiation lead to black hole evaporation / reduction of black hole mass?

Unruh and Hawking effects are similar but not quite the same. In particular, a freely falling observer far from the black hole does detect Hawking radiation. An inertial observer far from the black hole could be momentarily at rest relative to say Schwarzschild coordinates (for a simple black hole) and they would detect the radiation. They would still detect it as they begin to gently fall in the direction of the black hole. The black hole has interacted with the electromagnetic field and emitted photons for all to see, just like a star (well the process is not like a star, but the end result is, for observers far away).

The observer freely falling near to the black hole may or may not be able to detect the radiation, depending on the size of his detector and the length of time he has to register the effect. He hasn't got much time to look before he enters the horizon, so none of his measurements are going to be precise. In fact his energy measurements will be imprecise by an amount of the order of the Hawking temperature. And if he tries to measure for a time long enough to get that sort of precision, then he can't help but notice that spacetime is not flat. That is, his location will have moved through a distance of the order of the Schwarzschild radius. That is one way to see why such an observer's observations are not the same as those of an inertial observer in flat spacetime. (The equivalence principle only applies in the limit of small regions of spacetime of course). The other way of seeing it is to note that the electromagnetic field tensor in the Hawking case is not the same as in the Unruh case.

I am going to try to explain this without using any equations. I hope I don't make any mistakes along the way.

A freely falling observer "close" to the black hole horizon will not detect any Hawking radiation, like a non-accelerating observer in Minkowski space. This does not mean that any freely falling observer cannot detect Hawking radiation. If that was the case then even the observers which are at rest at spatial infinity (and hence freely falling to the gravity of the black hole) were not able to detect Hawking radiation. Even though freely falling observers which are close to the horizon will locally see it flat, if they are asked to extend their reach to feel the curvature of the horizon, they will feel the geodesic deviation as an acceleration. (This means that we cannot have a solid extended freely falling observer near the horizon.) Therefore one might say that if they try to look at the black hole as a whole then they will indeed see that it radiates.

To put in slightly different words, if near the horizon observer uses a detector vast enough to feel the curvature of the horizon, then that detector will feel acceleration. This means vast enough detectors can see the radiation. Also, the detector should be a non-accelerating object in the reference frame of the observer. Otherwise it will not work correctly, for example, in the case of flat space-time.