Cassandra multi-DC:write on LOCAL and read from any DC
The data is always written to all DCs, the LOCAL consistency levels are just saying that confirmation should come from the local nodes, not from other DCs...
From the DSE Architecture guide (also look to image there):

In multiple datacenter deployments, DataStax Enterprise (DSE) optimizes write performance by choosing one coordinator node. The coordinator node contacted by the client application forwards the write request to one replica in each of the other datacenters, with a special tag to forward the write to the other local replicas.
If the write write consistency levels is LOCAL_ONE or LOCAL_QUORUM, only the nodes in the same datacenter as the coordinator node must respond to the client request for the request to succeed. Use either LOCAL_ONE or LOCAL_QUORUM to reduce geographical latency and lessen the impact on response times of client write requests.

P.S. I recommend to read this guide to understand how Cassandra/DSE works - it makes things much simpler if you have this information.