Dear Lapey,
Can. 903 A priest is to be permitted to celebrate the Eucharist, even if he is not known to the rector of the church, provided either that he presents commendatory letters, not more than a year old, from his own Ordinary or Superior, or that it can be prudently judged that he is not debarred from celebrating. What is prudently judged is decided by the local ordinary, therefore the bishop is responsible for determining this. Therefore notifying the bishop is well, required.
[Again, this is not for a parish, but a private mass in a home]Agreed Mass at home is different, but still under the pastor and bishops’ control.
Can. 904 Remembering always that in the mystery of the Eucharistic Sacrifice the work of redemption is continually being carried out, priests are to celebrate frequently. Indeed, daily celebration is earnestly recommended, because, even if it should not be possible to have the faithful present, it is an action of Christ and of the Church in which priests fulfill their principal role. He doesn’t need a congregation at someone’s home to say Mass. He can celebrate Mass in his office or hotel room alone. I’ll address that in a second.
Can. 906 A priest may not celebrate the Eucharistic Sacrifice without the participation of at least one of the faithful, unless there is a good and reasonable cause for doing so. This would be at the discretion of the priest of a “good and reasonable cause”. And again no requirement to notify anyone because there are no other souls present. If there were then the person responsible for them would need to approve, as in the pastor. The hierarchy of a given location matters, bishop and pastor. They are charged with the responsibility of the souls charged to them, this is the reason for the permission.
I am interested in learning what is truly permitted, so if you can present your source if it is other than what I posted, please let us know.