You'll have to forgive my obtuseness. I only ever log into my Oracle servers as "oracle" and am able to login locally with
$ sqlplus / as sysdba
$ sqlplus dts
So today I was a little scared when I couldn't do this from an OS user other than oracle. The $ORACLE_SID variable was indeed set correctly. After quite a bit of hot Google action it turns out that sqlplus still uses the TWO_TASK environment variable. For some reason I had assumed it was deprecated and/or obsolete since I had never had to use it.
Sometimes it helps to step down from the mountaintop for a clearer view.