Make the decision after you see a doctor, and
Base the decision on whether there is an effective treatment available.
In other words, if it turns out that you have influenza, where early treatment (even before symptoms appear) can make a strong difference in outcome, then email. If it turns out that you have a cold, which will last a week if you treat it and seven days if you don't, then just let it go.
I actually just experienced this from both the warnee and warner position, though luckily not with students. A day after my parents left our house after the holiday, my mother called to say that she had been diagnosed with influenza A (despite having had the flu shot). When my toddler spiked a fever the day after that, I took it more seriously than I might have otherwise (we'd also all had the vaccine); it turned out my child also tested positive for influenza A, and the whole family was put on a prophylactic dose of antiviral medication. By that point I was also symptomatic, and accordingly informed colleagues with whom I'd had contact. Fortunately I'd already been keeping my distance from them before I was symptomatic, due to the previous notice, and none of them were affected. And thanks to our prompt treatment, no one else in our household came down with the flu, and my symptoms only lasted a few days.
On the other hand, colds around here are absolutely inescapable, and along with most of the folks around me I pretty much assume that anyone I come in contact with could be contagious between September and May. I'd be somewhat baffled to receive an email "warning" me that someone had come down with one.