This POV is the common one in discussions regarding the philosophy or foundations of statistics.
The model where an external skeptic has a more challenging prior has been taken for granted in that if you think about 2 scenarios – you flip a coin and verify heads vs. someone flips a coin and reports heads, the latter probability should be less than or equal to yours. So it seems rational to discount reports from external agents, (unless you question your own perceptions!). Therefore, it is reasonable for an external agent to discount your reports as well, and your design takes that into account.
When these scenarios are modeled formally, we end up with 1 person games (experiments to learn) and 2 player games (experiments to “prove”).
The following papers by Kadane and Seidenfield, then Jayanta Ghosh go into detail from a Bayesian perspective, which might help answer your question.
Randomization in the Bayesian Perspective. (link)
The Role of Randomization in Bayesian Analysis (pdf)