Triggers are not executed on the slave under row-based replication. However they are are executed on the slave under statement-based replication. Instead, when using row-based replication, the changes caused by executing the trigger on the master are applied on the slave.

This behavior is by design. The reason for this is that, if both the master and the slave applied the changes from the master and, in addition, the trigger causing these changes were applied on the slave, the changes would in effect be applied twice on the slave, leading to different data on the master and the slave.

If you wish for triggers to execute on both the master and the slave — perhaps because you have different triggers on the master and slave — then you must use statement-based replication. However, it is not necessary to use statement-based replication exclusively if you want to enable slave-side triggers; it is sufficient in such cases to switch to statement-based replication only for those statements where you want this effect, and to use row-based replication the rest of the time.