In the typical RPG model, players roll dice for antagonists, and GMs roll for everything else. Some systems have options in place to either reduce or remove the need for the GM to roll dice. If a PC attacks, the player rolls to hit; if a PC is attacked, the player rolls to defend. This is player facing. Two systems come to mind for this: Cinematic Unisystem and Ubiquity. They go about it in slightly different ways.
Cinematic Unisystem, the engine behind Eden Studios' licensed properties like Army of Darkness and Buffy the Vampire Slayer, breaks the game's stats down into three scores (by combining stats and adding an average die roll), which provide the check number and target number for anything affecting the NPC in question. To paraphrase my friend Thomas Deeny, the GM and the players are essentially playing two different games. The method Cinematic Unisystem uses could be adapted to a lot of games. In fact, D&D 4E and Star Wars Saga Edition use a similar philosophy to come to its Defense numbers (which replaced the original d20 Saving Throws)--attribute bonuses plus average die roll.
Ubiquity uses a dice pool system with very simple math behind it--each die has a 50% chance of success--so you know if your dice pool is 10, for instance, your average roll will bring five successes. Since the core mechanic is opposed rolls, it's easy to see how a GM would never have to roll. Even if there's an odd number of dice (in which case, Ubiquity rules you just roll one die to find out which way it goes), the GM can simply choose to round in a direction based on the situation--it's basically one more way a GM can set difficulty.
Not all systems can be made to work this way without extensive modification. For instance, no system with exploding dice could be modified to work this way easily. It would just make for too many wonky situations. Systems with no opposed roll mechanics--or at least target numbers derived from theoretical opposed rolls--are also off the table if one is looking for easy conversion to this resolution method.
I'm a fan of player-facing rolls because I find dice-rolling more enjoyable as a player than as a GM. Also, it really does help a system get out of the way of a story and gives the GM much more control over the pacing at the table. Few things slow a table down like waiting for the GM to roll for a bunch of attacks or defenses. And in the case of games with an opposed roll mechanic, it cuts the total number of dice rolls in half. Finally, games without the wild factor of GM rolls tend to be easier to prepare for.
If your system of choice lends itself to this type of change, or if you have a chance to try a new system, I recommend giving player-facing rolls a try.