There's always going to be standards like IEC 60601-2-47 which are difficult, rarely requested and subjective, so it's not economical for third party labs to be set up for. As such, modern regulations will never insist that accredited testing is required. However, agencies (e.g. Notified Bodies, auditors) may imply that third party testing is required, but really that's just a preference to make their life easier. If it's accredited, they can just record the report number, testing agency etc and move on. If it's not, they (should) check the detail in the reports, qualifications of persons doing, equipment is calibrated, raw data, completeness etc etc etc, which is a lot of work.
Some regulations (Brazil? China?) might insist on third party testing. However, this would be lip service only. Most labs are not set up so if a report existed it would be some form of a fudge, or detail would show the lab didn't really do the testing.