A responsible position is a post where the individual is expected to work without supervision or a higher authority, usually for a political purpose. This is in contrast to a job or police/military post, where there is a superior officer to answer to and who holds the right to direct work. For example, a shop steward and a member of parliament are in responsible positions; the former is responsible to the workers, the latter to the electorate.
Working in a responsible position may be unpaid, or if there is some compensation, it may not be considered a taxable salary. For example, the compensation of an MP is tax-free.
The related term position of trust refers to any kind of position, whether supervisory or not.
References
Responsible position Wikipedia(Text) CC BY-SA