In the U.S., as far as I know, it's all voluntary, and the rules/laws depend on the state in which you live. In my state, you're asked when you renew your driver's license if you want to be an organ donor. I imagine that even if you do not have a driver's license (and you are an adult over 18 years of age), you can still sign up for the organ donation registry. The driver's license renewal is just an easy way to do it. And, I believe that unless it was expressly against your wishes, your spouse or other relative could agree to donate your organs even if you had never signed up for the registtry. For children under 18, it would be their parents who would make the decision about organ donation since they are minors. However, permission is needed from someone, whether it was your original approval or your family's approval.
As for "common sense," there are indeed some religions (and non-religious individuals with deeply held beliefs) who are against organ donation. Which religions and for what reasons I'm not certain, but that's why you "opt-in" in the United States. Again, a lot of this depends by state laws and initiatives.