Inside the US, the required registration process varies from state to state and it is best to check with state's Dept. of Non-Public Education, or the appropriate dept. in your state. is probably not be a good idea to ask your local school district; often they don't know, and they may give you misleading information.
About.com homeschooling has a list of homeschooling requirements by state. Ann Zeise at A to Z's Home Cool Homeschooling also has a complete listing of resources state by state. I noticed that not all of these links are still live, so if someone else has a better list, please post it!
In NC, homeschools are legally treated like private schools, so each subsequent child can be registered to the existing homeschool. According to the NCDNPE:
As long as the home school is currently registered with DNPE; operating in compliance with the state home school laws; and, the student to be added lives in that household, DNPE does not need to be notified of the addition.
Student attendance, enrollment and academic records are to be kept and maintained by the home school -- not by DNPE.