Sharded MongoDB backup solution
You need to be a little more clear about your requirements:


Do you want to keep all 12 databases intact (12 instances versus all 12 in a single instance)? 
Are you intending to serve any queries from the backup servers?
Do you want multiple snapshots?


You can approach this from many different ways, but I will cover the most common approaches:

Put secondaries in US-West

This approach means you have a full, complete copy of your databases on the west coast.  You can make them hidden and priority 0 to make sure they don't serve slaveOK reads or become primary by accident.  To make them primary (in a disaster recovery scenario) you will need to re-configure the set manually if they are hidden and priority 0.

This is quite an easy approach, though it could be expensive.  You can run multiple mongod processes on a single host but they will be competing for resources and it will still have to be fast enough to keep up with the primaries on the east coast.

Use EBS snapshots

Assuming you are running with journaling enabled, you can just take an EBS snapshot of your data folder (dbpath) and you can back that up to S3 or use it to "seed" hosts on the west coast.

Mongodump

mongodump can be used to dump out all of your data from the Eastern hosts to the Western hosts.  However, the output (BSON files) will need to be restored before it can be used.  

Generally I would recommend a combination of the first two in order to combine the best resilience options with the most flexibility.