Are there illnesses that can be passed from pokemon to people and/or vice versa?

Yes, of course. In fact, many diseases can pass from one species to the next—even from pokémon to animals besides humans and vice versa—and a number of these have had historically devastating consequences.

While I could name a number of those (certain autoimmune diseases, smallpox, a number of neurological diseases including rabies, and even some forms of tuberculosis, to name a few), the most readily apparent, the most important, and the least graphic to talk about would likely be influenza. Technically, it’s just as rare for the flu to jump from a pokémon host to a human as it is for it to jump from animals to people, but it still is a possibility, especially in areas where consuming pokémon is prevalent and where sanitation isn’t quite up to code. Once it makes the jump, however, this strain of the flu can be just as contagious as any other, which means it can still be a very real threat. (You may have heard of the torchic flu outbreak that began in Orre a few years ago, for example.) So even if you do live in a region where pokémon consumption is low and where medical and sanitary education is high, it’s still important to vaccinate both yourself and your pokémon (if both you and your pokémon are medically capable of doing so), just in case you encounter a trainer or other traveler who isn’t quite as lucky as you are.

Of course, there are plenty of other diseases that don’t necessarily make the pokémon host sick but can certainly sicken you. This is especially true with parasites, as well as bacteria normally contained in pokémon fecal matter. Some pokémon—particularly reptilian ones—may also carry salmonella on their skin, so in general, it’s an excellent idea to wash your hands after cleaning your pokémon’s enclosures and after handling reptilian pokémon.

As for pokémon getting sick from our diseases, despite the fact that human beings are not actually the cleanest organisms in existence (there is a reason why we have an extensive list of procedures for self-maintenance), it’s less common for a disease to jump from us to them. However, it still happens, and it’s often said that if a trainer is sick, then their team is too. So always practice good hygiene on your travels (even if it doesn’t seem that convenient to you), and if you feel under the weather, be sure to keep your pokémon within their poké balls as much as possible and seek shelter to rest until you’re well again.