Good question. It's almost an art form sometimes. With any luck, the report gets to the SAR agency responsible. Often the original reporting party has disappeared or, worse, has no contact phone number. So then you've got to figure out if you've got enough information to be concerned and send someone to check it out. This depends on the experience of the SAR person -- often just the classic 'gut feeling.'
The best clue is the person says "I need help" or they've got some obvious and (preferably) dramatic injury. Otherwise, you've got to figure out what the reporting party saw, why they were concerned and what their experience/training is for that concern. The key here is tracking down the original RP and doing a good interview. The more people a report goes through, the less you can trust it (ok, duh!). It sounds like that might have happened to you -- too many synapses involved after you were talked to.
There's a few ranger supervisors now whose first question is "did they ask for help?" If not, then nothing is initiated. I'm often not in that camp because there's quite a few incidents where people don't really know they're in trouble and the reporting party is right in their concern. Getting to them early makes our lives easier later on.
A few years back, there was a guy reported in trouble on Whitney. It was a garbled cell phone report which cut off. I went up to check and got to Trail Crest during a bad lightning storm at about 5PM. I asked a guy coming down if there was anyone else up there. He said there was, but he'd talked to him and he seemed fine. Reassured and with the storm, I scurried back down the west side. The next morning, I headed up again to make sure. I found a guy on the switchbacks. When I first talked to him, he seemed fine -- for about 2 minutes when he started talking about his wife's garage sale on the summit. That's called a clue.
So I really don't know what the lesson is. Ultimately, we have to respond to more semi-bogus calls just to make sure we get most of the legitimate ones. Also that you really have to trust your experience and feel about something (whether you're the reporting party or a SAR person). Probably wrong more often than right.
Finally, as everyone here knows, once a person steps foot on the trail, it's really all on them. There's people who will help and SAR groups will do everything they can to get to them, but all responsibility is on them -- both to get themselves out if at all possible or to get word that they have a legitimate need for help. It's really darned amazing we do as well as we do.
None of the views expressed here in any way represent those of the unidentified agency that I work for or, often, reality. It's just me, fired up by coffee and powerful prose.