A while back I was on T4 only and I asked for a second opinion. I chose an endo who also worked privately with a leading thyroid specialist, as I thought he'd be more understanding and know up to date information.
Unfortunately he was very brusk with me, told me it was nothing to do with my thyroid as my TSH was already partially suppressed. I was being shown the door after 30 seconds. He told me none of my symptoms were thyroid related (they have cleared up 90% now that I'm on T3 only).
I stayed and fought for an increase of my medication, which I didn't get but did, at the last minute, I got a trial of T3. I was grateful to get the trial of T3 but afterwards realised that he never told me T3 existed or was an option. Initially I was being shown the door without being told about T3. I only got it because I asked for a trial of it.
I can't imagine he would have been so rude with a private patient and he may have told them about T3 rather than keeping it a secret from me on the NHS.
What I learned is that if I'm choosing to see a specialist on the NHS, choosing one who works in the private sector doesn't mean they're any better informed or will treat you any better (even if they work alongside top private specialists). My current NHS endo is good and I got her name from a recommendation from a local thyroid group.