A little bit of caution is needed when approaching this subject as there are several surfaces in cosmological solutions that can be interpreted (or misinterpreted in some cases) as horizons. Not all of them appear in all comsological solutions and in some cosmological solutions some of these surfaces are only present for part of the evolution of the solution.
If you were to mean the particle horizon, which is the limit of the observable Universe. The cosmological red shift at the particle horizon is always infinite, when there is a particle horizon. No matter can ever leave the particle horizon (regardless of the cosmological model, as long as there is a particle horizon) as it expands radially outwards at a local speed of c relative to any matter on the horizon (of course to us it expands even faster than c as it is also receding due to expansion).
An ever-expanding Universe with a cosmological constant approaches a state similiar to de Sitter space in late times, however de Sitter space itself is not a realistic cosmological solution as it is devoid of matter. De Sitter space does not have a particle horizon and for that matter it doesn't have any singularities either.
In de Sitter space the cosmic event horizon, which is the surface beyond which events happening now cannot affect us in the future providing we stay at the same spot (this should not be confused with the particle horizon), sits at a constant distant to the observer that is inversely proportional to the square root of the cosmological constant.
In answer to your question then, de Sitter space is symmetric in time and space which actually gives rise to seemingly different physical interpretations of it. However the black hole information paradox is specifically the loss of unitariness in the time evolution of a black hole and you have not justified why you think an equivalent thing should happen in de Sitter space. The loss of unitariness in actually arises because the event horizon disappears, but (naively at least) the information behind it also disappears. In de Sitter space the event horizon remains and even if it were to disappear by the cosmological constant miraculously going to zero we would still find the information behind it, so there is no analog to the information paradox in de Sitter space.
To take another example, consider Rindler coordinates in Minkowski spacetime, there is a horizon (the Rindler horizon) and there is radiation (Unruh radiation), however there is no information paradox as we know that unitariness is preserved in Minkowski spacetime and when the Rindler horizon is made to disappear by selecting a different coordinate system, all the information that was behind the horizon is still there.