The short answer is yes. However, the good news is that it is NOT very common. And better yet, the benefits we receive from owning pets far outweigh this very slight downside.
Diseases that can be transmitted from pets to people are called Zoonotic diseases. And they range from mild to quite serious. The best known and most feared zoonotic disease is rabies.
There are many others too - more than 200. They include ringworm, leptospirosis, mites, giardia, and salmonella (usually from reptiles). Although these sound scary, contracting one of these is very uncommon, and it's easy to reduce or even eliminate your risk of contracting a zoonotic disease with these simple tips.
Wash your hands after handling any animal, especially reptiles and birds.
Don't put off a vet visit if your pet shows any signs of illness.
Keep your pet clean! Bath time should be a regular activity. If you go to a dog park, wipe Fido down with a damp towel or doggie wipes from the pet store when you get home.
Deworm your puppy or kitten. Get your dog checked for worms by your vet every year.
Make sure your dog is getting Heartworm medication regularly; many of them are broad-spectrum and prevent other parasites too.
Pick up and dispose of your dog's poop on walks and in your yard immediately. Clean your cat's litter box every day.
Wash your pet dishes frequently.
Use flea and tick medication on your pet.
Wash any animal scratches and visit your doctor if an animal bites you.
So don't worry, follow these tips, and both you and your pet will stay healthy and happy.