These competencies are usually determined prior to the regular session, usually during character creation (some in-play creation options like HeroQuest's or FATE's notwithstanding). Once they're mechanically set, during play, what changes is only the determination of the difficulty of a task at hand, with the roll potentially implying whether the competency extends to this particular task or not. (And then there's infrequent character advancement during or between sessions.)
Now, let's say we skip the character creation parts that relate to competence. We don't spend skill points. We don't assign mechanical abilities (those with specified, numberical effects). We don't, in other words, numerically quantify a character's competence in any particular field. But we might write up a background, write down some things that are important about the character and might imply competence ("ex-soldier; artist; race car driver"), just without numbers.
Now, during play, whenever a task comes up, the GM just turns to the player and asks: How likely is your character to succeed at this? Let's say the player picks a number between 1 and 9, then rolls a D10 to meet or get under the number to succeed.
Instead of working on pre-determined general competence and GM-set difficulties, we let the players assess their characters' competence on the fly.
Here's the question: Would you have a problem with this approach, and if so, what is it and why do you think it's problematic?
Bonus question: Would your response change depending on whether you're playing with long-time playmates, new friends, or random strangers at a con?