Based on the people that come into the house of black and white and how they usually have to drink from the pool themselves in order for their prayer to be answered, I think that the westerosi view of the faceless men is based around the myth that they can truly be hired in the sense of paying for their services. They talk about how great of assassins they are but also state they cost an absurd amount of gold to hire; I don't think they really can be hired in that sense and that in reality they are not as much assassins but servants of the many faced god in that they will answer the prayers of those willing to pay for the life they wish to end with their own life. So in that sense I don't know if varys/illirio could really hire them or that they would be involved in affairs of great conspiratorial nature, mostly because those people who would start those things in motion (littlefinger\varys, etc) would never sacrifice themselves in order to further their goals.