When a religious community enters a diocese to establish an apostolate, they do so only because the local bishop has invited them to serve in his diocese. That's because canon law gives a bishop overall responsibility for the care of souls, administration of sacraments, exercise of worship, and works of he apostolate in his diocese.
A religious community that has pontifical right (i.e. the Vatican has approved their constitution) ultimately comes under the Vatican's authority. Nevertheless, the local bishops still holds authority over matters affecting faith and morals in his diocese, and can insist that religious living and serving in his diocese observe their communities' rules and the universal Church's rules and laws.
If a diocesan priest disobeys a Church rule, the bishop can deal with the matter directly. But is a religious community disobeys such a rule, the bishop must refer the matter first to the general superior of the community, and then, if necessary, to the Vatican's Sacred Congregation for Institutes of Consecrated Life and Societies of Apostolic Life.
A religious community doesn't need to check with the local bishop to assign a member to a ministry not usually associated with the order (e.g. a teaching community wishes to train a woman to be a nurse.)
- Little White Book, Diocese of Saginaw