Not every design project transfers full ownership — some are structured as a license instead, and the difference matters for how freely you can use and modify the work later.
Full ownership (copyright transfer)
The most common and generally most desirable arrangement for a custom logo or identity: full copyright transfers to you on final payment, meaning you can use, modify, and control the work without restriction, and the designer retains no ongoing rights to it.
Exclusive license
You get sole rights to use the work within agreed terms, but the designer technically retains underlying copyright. In practice this can function similarly to ownership for your purposes, but the specifics of what you can and can't do (modify it yourself, for instance) should be spelled out explicitly.
Non-exclusive license
Common with template or stock-based design assets — you're granted the right to use the work, but the creator can license the same or similar work to others as well. This is generally not appropriate for a custom logo meant to be unique to your business.
What to confirm before you sign
For a custom logo and brand identity, insist on full copyright transfer rather than a license, and get it explicitly stated in your contract. A license arrangement is more appropriate for things like stock photography or template assets, not your core brand mark.
The short version
For a custom logo, full ownership (not a license) is what you want — confirm the contract explicitly transfers copyright to you, rather than merely licensing you the right to use it.