Why? Don't you trust your DM?
Trust has nothing to do with it, this is a matter of design at the system level. When I say "objective skill system" I'm advocating for a design where the skill system is primarily player facing. It should be a tool players use to resolve situations; the actions you can take with a given skill should be encoded into the skill ahead of time and the gameplay should involve finding uses for those actions to get what you want.
There's a whole understanding of what skills are in that model, where a given skill modifier means a PC has access to a variety of abilities, and will look for opportunities to use them.
PF2 has two different sets of skill DCs. As others have said, for each skill there is a list of example skill DCs based on various tasks. Some Skill DCs are fixed, like needing a 15 to Treat Wounds.
In the sort of system I'm calling for, this is the all of skill checks. Some uses that aren't clearly codified might be worked out by analogy to the closest such task, but the design should be complete enough to make such situations rare.
Other skill DCs are tied to level - the level of an item, the skill DC of an adversary, that sort of thing. So it's a lot easier to Trip a guy who is 3 or 4 levels lower than you are, and harder against a guy who is higher level than you are. There's a lot more nuance than that, but that's the gist.
This is reasonable for opposed checks when you have a trait that can be referenced in a specific person, and I could even see the case for a level reference for crafting or as a derived trait for say, traps or location-bound spells, that kind of thing. It's a problem when abstract elements begin to have levels that aren't clearly defined, like a Level 15 climbing challenge, vs. a Level 13 climbing challenge.
You still need the DM to give you the DC. Even if you think you can swim across a pond at DC10, you might find some surprises once you get started. The DM is there to help adjust any static of example DCs, depending on the situation.
This is the thing I would like to be different. Skill DCs should be oriented toward player abilities, not toward DMs. The player should know that it's a DC 10 to swim across a calm body of water at Y speed, a DC 15 for a rough one, and so on, which then translated to with say a +10 bonus, a 100% chance of success at the former, and an 80% chance of success at the latter. Swimming is a PC ability that's deployed, potentially with a risk of failure associated, maybe some extra modifiers for speed adjustments and/or a take 10/20 system if the player is willing to spend time and isn't under threat.
To put it another way, I expect PCs to be declaring actions drawn from checks, not the GM calling for checks in response to PC action declarations. It's not "I want to climb that" "okay give me an Athletics check" it's the PC asking about the situation, and deciding to deploy the climbing action knowing the resolution mechanics ahead of time.
I bet you really don't like "secret checks".
I don't really see an issue if there's a chance of failure and a reason to hide the information from the PCs. I'd prefer most perception type abilities to be reworked as defenses, for example, that stealth is rolled as an attack against, which can be done entirely secretly. I do something similar with knowledge checks, rolling the "obscurity" of information against the PCs when they encounter stuff they might know about, and then providing the information upfront or not, without informing them of the roll results.
