Can a dog be majorly sick without showing signs and symptoms?

We purchased a pet today, and the moment we brought it home it defecated. Hanging from it's rectum was a worm, so we gave it some de-wormer. However, they want to take it JUST to get the shots tomorrow, not a physical check-up. I have another dog that I absolutely LOVE, and do not want her to get sick at all. So I'm asking: Can the dog who defecated have other sickness/symptoms without showing signs? It seemed lively and somewhat playful..and ate a lot of food the moment we got home.