From what I've read so far, neither.
There's another similar question that asks whether, in a true REST-style setup, the Application Server or the Database Server should hold the sessions. Several people responded that, while it's sometimes faster or more convenient to keep on the application server (or in your case the API server), it's preferable to store sessions in a database or Memcache.
If you keep them on the API Server, you run the risk of losing the sessions if the server crashes.
Also, (I paraphrase from AJ) keeping your sessions on the API server will only work if:
- Your clients always connect to the same server (aka "session affinity")
- Your server nodes all use a common mount point to spool sessions
I'm not sure the second one applies as much in this situation, but the database and/or Memcache (Thanks InspiredJW) idea does sound your best bet!
Thanks much