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In short? It depends, but common sense wins the day. Mild usually means good to go, moderate and beyond means no go.
In context, if you're having some mild allergic reactions or a slight runny nose with no other symptoms, you should be able to train anything-conditioning, strength work, plyometrics, mobility, etc. This is part of building the body to become anti-fragile. It may not be the best workout of your life, but you'll likely feel better for doing it. But what about the other end of the spectrum?
If you're experiencing more severe symptoms, including sinus pressure headaches, throat soreness, joint pain, etc. you probably shouldn't hit the gym. It's a sign that you're more than a little inflamed and need true rest, not stress-even hormetic (positive) stress, like exercise.
That said if you don't check the more severe symptom boxes and really feel like dislodging some mucus-try doing your own, nasty version of a purge, a little mobility or super light cardio/conditioning.
You could consider injecting a little citizen science and n=1 experimenting. Consider an informed test to monitor your athlete readiness (whether you're an athlete or not)-by testing H.R.V. (heart rate variability) or a full spectrum of tests on health monitoring systems like Omegawave or BIostrap. This is something we can easily do at home.
Something we all can relate to, but was first eloquently put into words by my guy Martin Berkhan is sickness behavior. Sickness behavior is defined as:
Sickness behavior is a coordinated set of adaptive behavioral changes that develop in ill individuals during the course of an infection. They usually (but not necessarily) accompany fever and aid in survival. Such illness responses include lethargy, depression, anxiety, malaise, loss of appetite, sleepiness, hyperalgesia, reduction in grooming, and failure to concentrate. Sickness behavior is a motivational state that reorganizes the organism's priorities to cope with infectious pathogens. It has been suggested as relevant to understanding depression, and some aspects of the suffering that occurs in cancer.
This promotes recovery and guards your immune system against additional insults. As Martin points out, if you're feeling tired, hurting, and anti-social, there you have it. Part of this can involve feeling weaker-this is because the activity of your CNS the central nervous system is down-regulated.
There is a potential for a lagging, hangover effect that can bleed into future training sessions. This is a lingering result of cytokines on the CNS. Don't concern yourself if you're not PR'ing upon your return to the gym, track, court, or field.
There you have it. My 2 cents? I don't believe in planning X workout on certain days (i.e. plyometrics EVERY Wednesday). To be perfectly clear, it's fine to do that but allow flexibility. If I'm dealing with something I feel like I'll fight off in a day or so, I may do a lighter session of conditioning (like a feel-good session on the concept2 rower) or rest entirely, only to hit my original targeted, more intense session later in the week.
When in doubt, respect your rest.