GameMasters are usually the ones handing rewards out, but there are certain games where players are asked to do so as well. The former may be used to adjudicating points and could conceivably fold "role-playing rewards" into the overall pool of rewards so players won't notice (but then, how is good role-playing incentivized?). The latter will often give everyone the same amount, for fear of hurting someone's feelings. Whether a player's role-playing is "good" or not is about as subjective as the criteria any given gamer uses. If we mean "acting", then the same players will get these awards over and over again, while a player whose strengths lie more in tactics will get passed over. If we mean whether or not a player did or said something that was "in character", I think we're getting closer to a more inclusive criterion, one available to all players. Depending on the tone of the game, role-playing rewards might also be given to anyone who does something "cool" or that makes the session more fun or interesting.
Regardless of how rewards are normally handed out, the best way to handle such rewards is at the moment they're triggered. Players can wait to the end of the session for their "objective" rewards, but if they just did something cool, just made a fun speech, just made a sacrifice in the name of "character", they should be rewarded right then and there. It's the equivalent of applause and a better incentive for players to role-play well than just another budget line at the end. By spotlighting key moments as "good role-play", the GM can "direct" the session towards an ideal, and players are less likely to be jealous of one another, having shared the moment with the GM, instead of seeing him or her like Dumbledore randomly throwing points Gryffendor's way. In-game, the sudden influx of points (especially when these are usable "hero points") mean a character's coolness gives them enough confidence to gain a bonus; if a sacrifice, then enough desperation.
In games where players are supposed to reward one another, you might allow them to nominate someone for a reward ("I think Joannie really deserves some hero points for that!"). In games that don't, the GM can still take cues from the table. If players all go "ooooh!" at one of their number's cool move, the latter should be rewarded. He or she has impressed. And don't be stingy. I personally love throwing points around (always more fun when there are associated tokens), and the free flow of points begets coolness. If the GM spends a lot of reward capital, players will be encouraged to do so too, using good role-playing to fuel more good role-playing.
The other anxiety-free alternative is not to give any points for "good role-playing" at all, but what fun is that?