Actually, this isn't really a rule change - it is just a change to how the rules are used at the table. D20 rolls affecting a player character are generally made by that player.
PC attack rolls and saving throws work as normal. NPC and monster attack rolls are flipped around though - instead of the GM rolling the d20 for the attack, the player rolls Evasion like a saving throw - d20 plus Evasion bonus. The DC for this roll is the monster's attack bonus plus 11. If the player makes the roll, the attack misses, if not, the attack hits.
When I re-introduce spells and supernatural abilities into playtest, the PCs will be using them on their enemies. Instead of the GM rolling saving throws for the monsters, the players will roll for the special attack, adding a d20 roll to their relevant bonus (save DC minus the basic 10, except where I introduce experimental rules). The DC for the roll will be the monster's save bonus plus 11. PC saves versus monster special attacks and spells will be as normal, except that the base DC will be 11 rather than 10. Spell resistance can be handled in a similar way.
Alert readers may be wondering why I'm adding 11 for the non-rolling party, rather than the official base 10. It is to ensure that the roll has exactly the same chance of success regardless of who rolls the die. For example, suppose the PC has attack bonus of +8 and the monster has Evasion bonus of +5. Adding 11 to the monster's Evasion bonus makes the DC for the roll 16, so the player requires a natural 8 to hit, giving a hit chance of 13 in 20. If the GM were to make the roll for some reason, the DC would be 11 plus the PC's attack bonus = 19, and the monster would be rolling d20 plus 5, requiring a 14 to evade the attack - again giving a hit chance of 13 in 20. If you were to reduce the non-roller's constant to 10, the hit chance would be 14 in 20 when the player made the roll but only 12 in 20 when the GM rolls. I suppose you could instead make the non-roller win in the event of a tie, which would even it back up again, but this would seem very weird to me having played for years rolling to equal the target number, rather than exceed it.
As mentioned above, the GM may wish to make certain rolls, for example to avoid giving the players information that they should not have. A compromise (which I haven't tried out for practicality yet) could be that the player rolls the die, but gets it to land where the GM can see it but the player can't.
This variant should make no difference to the outcomes of combats and other game events. It shifts some of the work of combat resolution to the players, helping the GM concentrate on her many other jobs, and helping to keep the players engaged between PC actions.