As Bryan says, there's only one relay, so if one side works and the other not, it's something else so don't waste your money on a new one!
The relay works on the basis that the load (current) generated by the bulbs causes the bi-metal strip to heat and bend, breaking the contact. The strip cools and bends back, remaking the contact. Pretty basic stuff compared to modern ECU-controlled bikes!
If the load is not high enough, the strip will not heat enough to bend and the flasher won't flash. This could be caused by a blown bulb or a bad connection. The latter is more often than not the case. Check the green (earth) connections to each indicator. In my experience this is the problem 99% of the time.
Good luck - let us know how you get on.
Assuming the O/S indicators come on (both of them) but don't flash