I'm skeptical of the value add when weighed against the risk of introducing unregulated / unverified medical tests with unknown false positive/negative rates during a crisis situation, but obviously folks disagree, and I'm sure I could be missing something.
This sort of thing is relatively easy to do for someone with the right background, but hard to do well. Aside from wasting reagents that could be potentially used by an actual diagnostic lab, some student running PCRs is going to get nowhere near acceptable test reliability. Those they falsely reassure will run around spreading it, and those they falsely diagnose will inappropriately self isolate. The more they test, the more accuracy matters.
Put another way (and apologies in advance for the example): if the guy was into watersports and noticed somebody's urine tastes sweet, is it a crime to tell them it might be diabetes? What if somebody asks him to taste theirs?
For anyone already feeling sick... stay home and self-quarantine.
If we could _just_ test everybody regardless of symptoms then we could quickly isolate people spreading it in ignorance. That is the main benefit I could reason out. Probably early detection and better planning as someone with it progresses, too.
The official statistics have a lag in them due to only showing confirmed cases.
Also depends on whether you're testing suspected cases or merely hypochondriacs.