It probably varies in different countries but in the UK the creator of any work owns that piece of work and all rights of use for it unless they assign those rights to some one else. Of course IP (intellectual property) law can be a quagmire and prooving who is the creator an absolute nightmare if it becomes neccesary.
Hence your own contract and terms should make it perfectly clear what rights you give the site owner on completion of the work. Do you hand it over outright and let them do what they wish once it's out of your hands? Or do you just give them a 'license' to use the design and code behind it and if so for how long? How do they change things if they do just want a minor change, a change of logo for instance, can they just go ahead or do they need to ask your permission?