Our depot CLI uses your local Docker credentials provider. So, any registry you've logged into with docker login or similar will be available when running a Depot build. This means that you can build images that use private registries like the example below.
FROM my-private-registry/project/image:version
...If you are experiencing issues with this, you should confirm you have logged into the registry from the machine where you are trying to run depot build. One way to ensure this is to try pulling the image via docker pull my-private-registry/project/image:version.