Usually yes the single electro-mechanical flasher relay is put in the left or right indictator circuit as selected. Normally you would have a three pin flasher relay with dedicated pilot terminal. See below for typical flasher wiring diagram for such.
![[Linked Image]](https://i.imgur.com/82dp2ii.jpeg)
As you can see the pilot light is fed by P terminal independent of selected flasher direction. You could nevertheless try swapping bulbs left to right to see if they draw differing current and note if fault switches sides.The other possibility is a two contact relay being used which would require diodes to common pilot light, one of which may be faulty.
More info on flasher needed.