Regardless of what the skill check is for - athletics, acrobatics, persuasion, etc - the DM will roll 1d10. The player will roll the same, plus their attribute.This way it's about the random will of the dice, but also incorporates who the character was designed to be. Your lucky gnome charmer is better at bartering than your grumbling dwarf hammerer.
If you want to make it more varied, you can use negative attribute scores, aka penalties, but ultimately I think that detracts from the fun and progression of play. Would you want to be the sourpuss with a negative charisma ATT? Or how about the scrawny, weakling with a negative STR that couldn't even climb a tree to snag an apple for lunch?
Let's say you're playing with a younger/newb audience and you as the DM don't want to decimate and discourage the fun by pwning them again and again. Sure a challenge is fun to overcome, but if dice/odds aren't weighted to favor advancing gameplay, I would daresay the players' interest will wane. I know mine would. As such, the DM could roll a d8 against the player's d10, or d6 against the player's d8, giving a slight advantage to the player's odds of success.
If you want to go with d20 rolls, and to give the oft neglected d12 a chance to get in on the action, the DM could roll a d12 against the players d20. In terms of favored odds, this would grant the players the highest advantage, which equals happy players and keeps the game moving.
Going back to a d10 system and for even odds, both the DM and player roll d10. If the DM gets a lousy roll and the players agree that it's too simple given the situation, then the DM will roll again (the higher of 2 rolls remains).
