Checking oracle can be very troublesome, but well worth it. I installed a mini nagios setup on the remote Solaris machines that are running oracle, and nagios uses nsca to report back the findings of the service checks against oracle. i.e. tablespace free and an host of others.
You are correct, you must have some client installed and most importantly, you need a correct environment to make your connection. Once you are able to make a DB connection from your nagios machine, then you can start making check_oracle checks.
Plus, you need a user account that has access to some tables in order to query tablespace free, and others.