According to the Splunk documentation1, in a search head cluster, the KV Store Primary is the same node as the search head cluster captain. The KV Store Primary is responsible for coordinating the replication of KV Store data across the cluster members. When any node receives a write request, the KV Store delegates the write to the KV Store Primary. The KV Store keeps the reads local, however. This ensures that the KV Store data is consistent and available across the cluster.
References:
* About the app key value store
* KV Store and search head clusters