I would think “private Mass” is possible, but unlikely on a consistent basis. That he is saying Mass as other churches of his diocese would be more likely in my opinion. After all, canon 389 provides: “He [the diocesan bishop] is to preside frequently over the celebration of the Eucharist in the cathedral church or in another church of his diocese, especially on holy days of obligation and other solemnities.”
Since canon 396 obliges him to certain episcopal visitation of his parishes, he may be off doing exactly that. Of course, he may also be off, administering confirmation personally, etc.
As the “foremost dispenser of the mysteries of God” in his diocese, it is improbable that he is customarily merely “in attendance” at Mass, but far more likely that he is celebrating it daily. The only reason otherwise that I could imagine would be poor health.
If your diocese has a website or newspaper, you may be able to find the bishop’s schedule. Most have daunting schedules, and perhaps his Mass whereabouts may be inferable from those sources.
Then too, cardinals, since they have Vatican duties and are subject to the call of the Supreme Pontiff, are often unable to be present in their dioceses on a consistently visible basis…
