Let's evaluate a few certain items here, and I wish to use the wedding photography business as a model, as it was previously used.

First and foremost, if you provide textual content, graphical content, video content or any other aspect for the web site, you obviously already own it (or should) and are thereby granting the designing firm a license to use it in conjunction with the services they are providing. A wedding photographer does not take content, then change the presentation, and take ownership as is often the case in a web development situation.

On the flip side, if you gave them a list of services you perform, or products you sell, and gave them the freedom to create the content (text, sell copy, and they take the product/service photos), then they very well do own the content at that point.

If you allow them to register your name as their own, rather than putting you as the domain registrant, then yes, they will own that as well. It is highly important that you "own" the domain name, for if you wish to ever move to another designer/hoster you will be unable to take your domain with you - which may be a very bad sitaution for your business.

Secondly, if they are programming a database to house your data, code to display that data, code to control user login, code to allow user interaction, all of the logic and code will be their property to use how they see fit. So they may make your site, and take the database design (not I did not say the actual data), the programming logic/code and re-use it on any number of other sites at will. You will never have any control over this, at least not any more control that you have over owning the code to Windows. So, all logic code is their property, not yours.

That is where the wedding photography example comes back into play, the photos are theirs because they created them using their method. In this case, the logic is the developers as it is there method, the data (or actual viewable content) of the web site is only theirs if you commissioned them to create it for you.

There is not an argument merely over the web site ownership, but arguments over domain owner ship, content ownership and logic ownership at this point.

If you allow them to register your domain name, you may end up not being the rightful owner to that name, so don't allow that.

If you allow them to generate the content rather than you providing it to them, they own it, thus owning the true "website" as the website is the content. So don't allow that, provide them with content that you personally generate and you will retain the rights to that content and they will be unable to use it on other sites, or an this site should you choose to host elsewhere.

If they have developed any logic code to display your data or their created data, that logic code is, and always will be their property.