Do black holes produce thermal radiation? Do black holes produce thermal radiation, as expected on theoretical grounds? Does this radiation contain information about their inner structure, as suggested by gauge–gravity duality, or not, as implied by Hawking's original calculation? If not, and black holes can evaporate away, what happens to the information stored in them (since quantum mechanics does not provide for the destruction of information)? Or does the radiation stop at some point leaving black hole remnants? Is there another way to probe their internal structure somehow, if such a structure even exists?
Hawking radiation is the black body radiation that black holes emit. It is a well established theoretical result but it has never been observed.  For all currently known astrophysical black holes it will be totally insignificant and will never be observed.  The reason for this is that the equivalent temperature of the black body is inversely proportional to the mass of e black hole.  For all currently known black holes the temperature would be a tiny fraction of a degree above absolute zero.

Quantum mechanics is a probabilistic approach to physics, and deals with all kinds of probabilities and uncertainties. There is one such uncertainty of time and energy, given by (Δt) (ΔE) ≥ ℏ/2

One of its predictions, which has been experimentally confirmed, is that a particle may take an energy E from “nothing” and yet not violate the Law of Conservation of mass-energy if it “returns” the energy within a time given by (Δt)≥ ℏ/(2E).

One of its many implications is that random vacuum fluctuations constantly take place everywhere, i.e. even if a certain region of space has nothing or is purely vacuum, it can still constantly generate and annihilate particles due to the E-t uncertainty.

fast forwarding

When such a fluctuation causes a particle and its anti-particle to be generated on the edge of the event horizon of a black hole, one of the two enters the black hole and the other escapes. Now for returning the energy to the system, the particle relies on crashing into its anti-particle causing an “annihilation” and conversion of mass to energy. In the scenario I described, this can never take place since the pair of particles have been separated effectively forever. The particle that enters the event horizon has been trapped in an infinite potential gravitational well, so can be thought to have negative energy. So anti-intuitively (wait, what part of all this is even intuitive!), the particle left lurking outside the absorbs some energy from the black hole, (thus conserving mass-energy) gets converted into a “classical” photon and escapes the vicinity of the black hole.

Thus little by little, such events cause a black hole to effectively “evaporate” and diminish. 

The apparant paradox about the fact that what happens to the information that goes into a black hole is a matter of debate. No one knows exactly how this Black Hole Information Paradox can be solved, however certain speculations are provided by Kip Thorne and Stephen Hawking, which include Firewall theory and Ads/CFT.