I think a Moon sentence should be just as easy for a PC to use as for the Opposition, since the GM gets to work with the invoked Sentence after it is used. Even if the players use a Sentence mechanically in one scene, the responsibility of the Moon's setting/storytelling moves back to the GM after the conflict, and they can use the previously invoked Sentence in that scene narratively, and later mechanically themselves, with it being a callback.
A Moon sentence is meant to broaden the Moon's sense of place, like a unique twist that could only really happen there, and that means the Moon Sentences should be to either side's benefit. Your example of "Secret police" is a double-edged sword waiting over everyone, since it is the Moon's police, not necessarily the opposition faction's police. The secret police could just as easily arrest an opposing agent as a PC, and that's part of the drama. But once a PC invokes a Sentence in the story, that's become part of the story now, and after the mechanical dust has settled, the GM can apply that sentence in the narrative in other ways the PCs can't. The secret police could question the PC after they arrest the Opposition, or start tracking the Band more closely as an interested party, or remain involved as an interested party with their own goals separate from either of the first two groups. Once the scene has changed, and when narratively compelling, the GM can invoke those same secret police Sentence, and while mechanically it might not be a solid move, it'll narratively have more punch.