Well, saliva (or the rate of production of saliva in your mouth) can tell you a lot about health, but the information you acquire is extremely general, so it won't really give you an entire health analysis in terms of diseases, sugar levels, body temperature, likewise.
As for what saliva does tell about your health, here is an extract from a source:
"Dr. Stuart Froum, president of the American Academy of Periodontists, says: “If the eye is the window to the soul, the mouth is the window to the body.” Saliva in the mouth is one of the best indicators of health. If something is off, there is a good chance your saliva will change as a direct reflection.
Saliva is produced around the clock and is affected not only by every bite of food you take, but also every other thing you put in your mouth. If you notice your saliva is too stringy or thick, your body might be having a hard time producing saliva. This could be because of the prescription and over-the-counter medication you are taking.
Studies indicate that an imbalance of the ‘stress hormone’ cortisol in saliva points to emotional sensitivity to stressful situations...the measurement of cortisol in saliva is becoming increasingly acceptable as a screening test for hypercortisolism, such as depression, hypertension and diabetes.