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Anyone is able to reply to the BMJ or write to a newspaper or express their opinion about anything online in a forum - perhaps such as this - be there layperson or medical professional. At what point does 'free speech' become something at odds with the organisation you work for/represent and something you can/should be disciplined about?
When writing as part of an organization you are acting as their agent and representing their views. I know people who have written technical blogs for a company and they were rightly careful to keep to the values of the company. This is about brand management for an organization. At the same time there was someone who wrote a non-company blog who was probably more controversial but they were clearly his views not the companies. The problem comes when weight is given to views because they come from a particular organization which is respected. To take the point a little further I would argue that information gained whilst working for an organization belongs to the organization hence an employee is for example not free to talk about future product plans this can even cause big problems with the SEC.
If I were running the Essex service I would see two issues
1) That Collins and Newton damaged the brand of the service
2) That Collins and Newton demonstrated a lack of awareness of a disease they were treating and insulted their patients.
The Essex service initially seemed concerned about the first point. Where as I think the second is the significant one (especially given the poor brand image of fatigue clinics!). The question is how do they act with patients. Do they have sufficient knowledge of the disease they are claiming an expertise in. Are they aware of current research or just the highly spun PACE trial results. An internal review should look at these issues. They should also look at the clinics processes if some staff are unaware of current research then are others. And why is there a culture in the clinic that means members of staff feel justified in making such comments publicly and privately.
I generally see these as management issues and it suggests that the management of the service is poor and hence those in charge should go.