Not necessarily. Faculties are necessary for witnessing marriages, hearing confessions, performing confirmations. One can hold faculties in multiple dioceses–some dioceses even allow priests who receive faculties within them to exercise those same faculties elsewhere in the same ecclesiastical province (e.g. if one has faculties in any diocese in the Province of Ohio, he has the same in every diocese therein).
The celebration (or concelebration) of Mass does not require a faculty, since faculties are necessary for the validity of the sacraments. Public celebration requires only the permission of the pastor of the parish. No recourse to the bishop is necessary.
-ACEGC