I believe that many people on this thread have so far answered from the point of view of their trade/MOC only.
For instance, Pusser, who is a logistics officer, did not have to learn single flag meanings save for those in the subset he needed to get his harbour watch-keeping qualification. Boatswains would only be required to learn the ones for their trade, such as the boat signals and the ones used in RAS and ceremonials. Etc.
Naval communicators and MARS officers, on the other hand, are required right from their first trade qualification course to learn and retain ALL the international and Naval single flag meanings.
By the time a MARS officer stands her own watch on an operational ship and by the time a Naval Communicator has completed his last trade qualification course, they have also learned the frequently used multiple flag international meanings and also the various important and frequent tactical signals without the need to look them up.
This said, the bridge naval communicator is nevertheless expected to always look up the meaning in the appropriate tactical publication, in case she gets a brain cramp and mixes two signals up, or even worse - in case the CO and the Yeoman chatting on the bridge wing get a whiff of the mistake and round up on the poor signalman. The "interpretation" of the signal reported to the OOW usually serves as a "check" on the OOW who has (most of the time) already decoded in her mind what was signalled, hence the importance of making sure by looking it up.
Don't if they still do this, but in my days (pre-dinausors extinction) when too many signals had been mixed up by junior officers and/or signalman in a squadron/division/sub-division, we used to provide extra training by carrying out OOW manoeuvres by flag only.