The question needs to be more specific. Most standard aviation policies in place at flying schools will cover the instructor's THIRD PARTY liabilities, i.e. legal liability to third parties such as damage caused to someone else's property or injuries to others. However it is usual in such policies for there to be a CREW EXCLUSION as far as PERSONAL INJURY OR DEATH is concerned. That means that the pilot's own injuries would not be compansated under that policy.
That may seem unfair but it is the norm in these policies. I think that it is because crew injury would normally be covered by an employer's liability policy, but of course that does leave the self-employed pilot exposed. Buying your own medical policy is the answer to this.
However, for the freelance instructor and examiner the situation is less clear and it is likely that you need to have your own professional liability insurance to cover the third party liabilities when you operate outside your normal flying school environment.
Hope this is useful.