How do the mint set the value? Surely it cannot be a value plucked from the sky.
Actually, that's exactly how it is set - plucked from the sky. Not by the mint, though, by Congress (in the case of the US, similarly in other countries). The US mint merely executes orders given by the US Congress and signed into law by the US President. For example, the wildly popular Silver Eagle bullion coin is authorized by the Statue of Liberty - Ellis Island Commemorative Coin Act. The act specifies the denomination ($1 in this case), the metal composition (99.9% pure silver, in this case) and the visual appearance. See description here.
ALL the US coins that are minted - are minted exactly as described and prescribed in the relevant legislation, and that includes the denomination.
Precious metal coins are not intended for circulation, but rather for collectors. Thus, the denomination doesn't correspond to the coins' real (intrinsic) value - the actual value of the metal is significantly higher than the denomination of the coin (with the notable exception of the Canadian $20 for $20 series).
Furthermore, what does the value imply to the bearer? Is it redeemable for $25 at any time?
Yes, they're legal tender. Obviously it would be ridiculous to do that since the metal content worth much more than the denominated value. It has also been set as a legal precedent that payment with these coins cannot be taken at face value, but rather at their real (metal content) value, for the purposes of income tax assessment.