Just to clarify, the FP player also gets a chance to respond to his own response action, and may do so before the Shadow player may. Similarly, the Shadow player can respond to his own response (but not until after the FP player gets a chance to).
For example:
1. Grishnakh kills a companion.
1A. The Free Peoples player may respond to this, but passes.
1B. The Shadow player plays
The Ring's Oppression to add a burden.
1B1. The Free Peoples player may respond to this, but passes.
1B2. The Shadow player plays
The Ring is Mine to add another burden.
1B2a. The Free Peoples player may respond to this, but passes.
1B2b. The Shadow player may respond to this, but passes.
1B3. The Free Peoples player may respond to the burden from
The Ring's Oppression, but passes.
1B4. The Shadow player may respond to the burden from
The Ring's Oppression, but passes.
1C. The Free Peoples player may respond to the companion being killed, but passes.
1D. The Shadow player may respond to the companion being killed, but passes.
2. The game continues.
Note that if the Shadow player had a second copy of
The Ring is Mine, he could have played it in response to either
The Ring's Oppression (in addition to the copy he already played) or in response to the first copy of
The Ring is Mine. If you have 4 copies of
The Ring is Mine, and a burden is added, you can play them all in response to the first burden, or play the first in response to the initial burden and each of the others in response to the previous copy of
The Ring is Mine, or anything in between.