It all depends on the screening tests. You have to have a general practitioner/internist you can trust, and take his/her advice. Don’t ask for something just because you saw it on the internet 🙂 Some tests, like blood tests, are trivial. Others, like CT scans, could add a bit to your lifetime radiation exposure, but are basically harmless. And others, like colonoscopies, are very invasive and have a risk of complications, and should be done only under a doctor’s advice that it’s necessary or recommended, especially for older age groups.
Take your MDs advice. If he/she recommends a test, get it done. If you don’t need it, don’t get it.