One doesn't even need to consider de sitter space to see this. Just take an accelerated observer in a flat space-time. For a person accelerating with acceleration $a,$ there is an event horizon $c^2/a$ distance away from him. Universe outside this horizon is oblivious to A. One could also consider following example, let's say at $t=0$ a light is shined towards a particle $c^2/a$ distance away. At $t=0,$ the particle also starts moving with acceleration $a$ but away from light. According to accelerating observer light never reaches him and the source of light is at the constant distance away at all time. According to someone not accelerating, the light keeps getting closer and closer to the particle and meets him asymptotically. [See [Hyperbolic Motion][1])
What is important in GR, or in fact in physics, are quantities called Dirac observables. For example, the distance between two particles is Dirac observable. It will not depend on coordinate chosen to describe it. Any quantity that is independent of gauge chosen is Dirac observable. 562
Time between two events is not a Dirac observable as it depends on coordinate chosen. So it boils down to determining what all quantities are Dirac observables.
In case of gravitational wave (GW), the effects of GW are Dirac observable. The distance between two particles oscillates when GW passes through them. Thus effects of GW can't be switched off by simply choosing different coordinates.
Tidal forces are another example of such quantities. If two neighboring geodesics diverge then it's a property of manifold. Changing chart can't change the fact that geodesics are diverging. If someone goes in the black hole, the tidal forces are for real. Just locally choosing a Minkowskian metric at each instant won't help. Mathematically speaking they won't form a coordinate basis. It is related to tetrad formalism. Tetrad formalism is helpful in telling how a particle sees his surrounding, but at the expense of all the beautiful result based on the coordinate basis. So one can't say that since I have the Minkowskian metric at every instant there should be no tidal force. That is only true if you use coordinate basis (because lie derivatives of coordinate basis are zero).
Hope this helps!
1[]: https://en.wikipedia.org/wiki/Hyperbolic_motion_(relativity) 562