When a design project ends, you own less than you might think, unless the contract says otherwise. The finished logo or website is yours to use, but the working files behind it, the copyright to the design, and sometimes the fonts and the code can all stay with the studio by default. That is not a scam. It is what happens when nobody spells it out, and it is the thing to settle before you sign, not after the last invoice clears.
Start with the difference between the finished thing and the thing that made it. A studio hands over the exports: the logo as a PNG and an SVG, the website live on your domain, the deck as a PDF. What it does not always hand over is the editable source, the layered logo file, the design file the site was built from, the original artwork. You can use the exports anywhere. You cannot change them, or take them to another designer to build on, without the source. If you want the ability to edit and extend later, that has to be named as a deliverable, because it is often treated as the studio's own working material and priced separately.
Then there is who owns the copyright, which is a separate thing from who holds the files. In a lot of places the person who makes a creative work owns the rights to it by default, even after you have paid, until those rights are transferred to you in writing. So you can pay in full for a logo and still not own the copyright unless the agreement assigns it to you. A good studio transfers the rights to the final identity as a matter of course and says so plainly. If your contract is silent on it, ask, because permission to use something is not the same as owning it, and the gap shows up the day you try to register your own logo as a trademark.
Fonts are the trap almost nobody checks. The typeface in your brand is rarely owned, it is licensed, and the licence has terms: how many people may use it, whether it covers your website as well as print, sometimes a fee that renews every year. When a studio picks a font for you, you inherit that licence and whatever it costs. Get it in writing which fonts your brand depends on, whether the licence is in your name or the studio's, and what it actually covers, or you can find out a year later that the font on your own site was never cleared to be there.
For anything built, code or a whole website, the question is whether you can pick it up and run it somewhere else. Custom code written for you should be handed over and hostable anywhere. A site built on a studio's own proprietary platform, or tied to their hosting, is a different arrangement: it works beautifully right up until you want to leave, and then you learn you were renting, not owning. Ask plainly whether you can move the site to another host or another team without a rebuild. If the answer is no, that is not automatically the wrong choice, but you should know you are making it.
Two small things people hand over without a thought and then regret. Your domain name and your hosting and analytics accounts should be registered in your name, with you as the owner, not sitting inside the studio's account with you as a guest. It is convenient to let the studio set everything up under their login, right up to the day you part ways and find the address your customers type belongs to someone else. Set those up in your own name from the start and give the studio access, rather than the other way round.
None of this needs a lawyer to fix. It needs one honest paragraph in the agreement before work starts, covering five things: which source files you receive, that the copyright in the final work transfers to you, which fonts are licensed and to whom, whether the build is yours to host anywhere, and that the domain and accounts are in your name. A studio worth hiring will have those answers ready and put them in writing without flinching, because it has had the conversation before. If a studio goes vague when you ask who owns what at the end, you have learned something useful about them while it still costs you nothing. Ask the question before you sign, not when you are trying to leave.