In these circumstances the driver does not have to wait for a green signal. That the signal may be adjacent to the driver, or across the intersection, is irrelevant: if the signal faces the driver the signal applies to the driver's actions.
A right turn is permitted in this situation, provided the driver comes to a full stop before proceeding, there are no pedestrians within an adjacent crosswalk, and no approaching vehicle constitutes an immediate hazard.
Note, however, that Subdivision (c) indicates that a driver may not make this right-turn when faced with a steady red arrow symbol. Subdivision (d) is a similar prohibition against pedestrians entering a crosswalk when facing a red signal.
[California Vehicle Code Section 21453 is the relevant authority in California. It reads:
(a) A driver facing a steady circular red signal alone shall stop at a marked limit line, but if none, before entering the crosswalk on the near side of the intersection or, if none, then before entering the intersection, and shall remain stopped until an indication to proceed is shown, except as provided in subdivision (b).
(b) Except when a sign is in place prohibiting a turn, a driver, after stopping as required by subdivision (a), facing a steady circular red signal, may turn right, or turn left from a one-way street onto a one-way street. A driver making that turn shall yield the right-of-way to pedestrians lawfully within an adjacent crosswalk and to any vehicle that has approached or is approaching so closely as to constitute an immediate hazard to the driver, and shall continue to yield the right-of-way to that vehicle until the driver can proceed with reasonable safety.
(c) A driver facing a steady red arrow signal shall not enter the intersection to make the movement indicated by the arrow and, unless entering the intersection to make a movement permitted by another signal, shall stop at a clearly marked limit line, but if none, before entering the crosswalk on the near side of the intersection, or if none, then before entering the intersection, and shall remain stopped until an indication permitting movement is shown.
(d) Unless otherwise directed by a pedestrian control signal as provided in Section 21456, a pedestrian facing a steady circular red or red arrow signal shall not enter the roadway.