when the idea is baked into how your whole traffic system operates, you don't have these issues. People turning right have the same right-of-way as people going straight through an intersection. People turning left must wait for all traffic to clear from the opposing lane before turning.

As for someone to your left going straight thru the intersection... you, turning right, must yield to those going straight. However, there is not always traffic coming straight through the light. If someone on the perpendicular street turns right instead of going straight thru, you may turn right from your lane into their lane because there is a pause in traffic flow coming through the intersection.

That probably doesn't make much sense but the bottom line is that when everyone knows the rules and everyone observes them, right on red is mostly painless.