Unless you're allergic to dogs or the dog has some bad disease then no you cannot get sick from sleeping with a dog.
I sleep with a big yellow lab dog all the time and never have I gotten sick and he loves sleeping in the same bed as me and I love having him sleep with me as well.
Many people sleep with dogs without any health problems and you'll only get sick from sleeping with a dog if they have some severe disease that you might catch or if you're allergic to them.
Other than that sleeping with dogs is very safe.