The registration requirements of Section 801 of the Food and Drug Administration Amendments Act (FDAAA 801) (PDF) and the International Committee of Medical Journal Editors (ICMJE) policy on Clinical Trial Registration under the ICMJE Recommendations do not exclude clinical studies with no external sources of funding ("unfunded" studies). See FDAAA 801 Requirements for more information on which trials must be registered under FDAAA 801.
In general, an unfunded study should be registered via the PRS account of the Sponsor. When an investigator is considered the Sponsor (a Sponsor-Investigator), the study should be registered using the PRS account of the investigator's affiliated institution with the Responsible Party indicated as Sponsor-Investigator. ClinicalTrials.gov will then display the investigator as the Sponsor instead of the investigator's institution. The Data Element Definitions page on ClinicalTrials.gov describes the related data elements for Responsible Party.